Theory AOT_PLM
1
2theory AOT_PLM
3 imports AOT_Axioms
4begin
5
6
7section‹The Deductive System PLM›
8
9
10unbundle AOT_no_atp
11
12AOT_theorem "modus-ponens": assumes ‹φ› and ‹φ → ψ› shows ‹ψ›
13 using assms by (simp add: AOT_sem_imp)
14lemmas MP = "modus-ponens"
15
16AOT_theorem "non-con-thm-thm": assumes ‹❙⊢⇩□ φ› shows ‹❙⊢ φ›
17 using assms by simp
18
19AOT_theorem "vdash-properties:1[1]": assumes ‹φ ∈ Λ› shows ‹❙⊢ φ›
20 using assms unfolding AOT_model_act_axiom_def by blast
21
22text‹Convenience attribute for instantiating modally-fragile axioms.›
23attribute_setup act_axiom_inst =
24 ‹Scan.succeed (Thm.rule_attribute [] (K (fn thm => thm RS @{thm "vdash-properties:1[1]"})))›
25 "Instantiate modally fragile axiom as modally fragile theorem."
26
27AOT_theorem "vdash-properties:1[2]": assumes ‹φ ∈ Λ⇩□› shows ‹❙⊢⇩□ φ›
28 using assms unfolding AOT_model_axiom_def by blast
29
30text‹Convenience attribute for instantiating modally-strict axioms.›
31attribute_setup axiom_inst =
32 ‹Scan.succeed (Thm.rule_attribute [] (K (fn thm => thm RS @{thm "vdash-properties:1[2]"})))›
33 "Instantiate axiom as theorem."
34
35text‹Convenience methods and theorem sets for applying "cqt:2".›
36method cqt_2_lambda_inst_prover = (fast intro: AOT_instance_of_cqt_2_intro)
37method "cqt:2[lambda]" = (rule "cqt:2[lambda]"[axiom_inst]; cqt_2_lambda_inst_prover)
38lemmas "cqt:2" = "cqt:2[const_var]"[axiom_inst] "cqt:2[lambda]"[axiom_inst] AOT_instance_of_cqt_2_intro
39method "cqt:2" = (safe intro!: "cqt:2")
40
41AOT_theorem "vdash-properties:3": assumes ‹❙⊢⇩□ φ› shows ‹Γ ❙⊢ φ›
42 using assms by blast
43
44AOT_theorem "vdash-properties:5": assumes ‹Γ⇩1 ❙⊢ φ› and ‹Γ⇩2 ❙⊢ φ → ψ› shows ‹Γ⇩1, Γ⇩2 ❙⊢ ψ›
45 using MP assms by blast
46
47AOT_theorem "vdash-properties:6": assumes ‹φ› and ‹φ → ψ› shows ‹ψ›
48 using MP assms by blast
49
50AOT_theorem "vdash-properties:8": assumes ‹Γ ❙⊢ φ› and ‹φ ❙⊢ ψ› shows ‹Γ ❙⊢ ψ›
51 using assms by argo
52
53AOT_theorem "vdash-properties:9": assumes ‹φ› shows ‹ψ → φ›
54 using MP "pl:1"[axiom_inst] assms by blast
55
56AOT_theorem "vdash-properties:10": assumes ‹φ → ψ› and ‹φ› shows ‹ψ›
57 using MP assms by blast
58lemmas "→E" = "vdash-properties:10"
59
60AOT_theorem "rule-gen": assumes ‹for arbitrary α: φ{α}› shows ‹∀α φ{α}›
61 using assms by (metis AOT_var_of_term_inverse AOT_sem_denotes AOT_sem_forall)
62lemmas GEN = "rule-gen"
63
64AOT_theorem "RN[prem]": assumes ‹Γ ❙⊢⇩□ φ› shows ‹□Γ ❙⊢⇩□ □φ›
65 by (meson AOT_sem_box assms image_iff)
66AOT_theorem RN: assumes ‹❙⊢⇩□ φ› shows ‹□φ›
67 using "RN[prem]" assms by blast
68
69AOT_axiom "df-rules-formulas[1]": assumes ‹φ ≡⇩d⇩f ψ› shows ‹φ → ψ›
70 using assms by (simp_all add: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp)
71AOT_axiom "df-rules-formulas[2]": assumes ‹φ ≡⇩d⇩f ψ› shows ‹ψ → φ›
72 using assms by (simp_all add: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp)
73
74AOT_theorem "df-rules-formulas[3]": assumes ‹φ ≡⇩d⇩f ψ› shows ‹φ → ψ›
75 using "df-rules-formulas[1]"[axiom_inst, OF assms].
76AOT_theorem "df-rules-formulas[4]": assumes ‹φ ≡⇩d⇩f ψ› shows ‹ψ → φ›
77 using "df-rules-formulas[2]"[axiom_inst, OF assms].
78
79
80AOT_axiom "df-rules-terms[1]":
81 assumes ‹τ{α⇩1...α⇩n} =⇩d⇩f σ{α⇩1...α⇩n}›
82 shows ‹(σ{τ⇩1...τ⇩n}↓ → τ{τ⇩1...τ⇩n} = σ{τ⇩1...τ⇩n}) & (¬σ{τ⇩1...τ⇩n}↓ → ¬τ{τ⇩1...τ⇩n}↓)›
83 using assms by (simp add: AOT_model_axiomI AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def)
84AOT_axiom "df-rules-terms[2]":
85 assumes ‹τ =⇩d⇩f σ›
86 shows ‹(σ↓ → τ = σ) & (¬σ↓ → ¬τ↓)›
87 by (metis "df-rules-terms[1]" case_unit_Unity assms)
88
89AOT_theorem "df-rules-terms[3]":
90 assumes ‹τ{α⇩1...α⇩n} =⇩d⇩f σ{α⇩1...α⇩n}›
91 shows ‹(σ{τ⇩1...τ⇩n}↓ → τ{τ⇩1...τ⇩n} = σ{τ⇩1...τ⇩n}) & (¬σ{τ⇩1...τ⇩n}↓ → ¬τ{τ⇩1...τ⇩n}↓)›
92 using "df-rules-terms[1]"[axiom_inst, OF assms].
93AOT_theorem "df-rules-terms[4]":
94 assumes ‹τ =⇩d⇩f σ›
95 shows ‹(σ↓ → τ = σ) & (¬σ↓ → ¬τ↓)›
96 using "df-rules-terms[2]"[axiom_inst, OF assms].
97
98
99AOT_theorem "if-p-then-p": ‹φ → φ›
100 by (meson "pl:1"[axiom_inst] "pl:2"[axiom_inst] MP)
101
102AOT_theorem "deduction-theorem": assumes ‹φ ❙⊢ ψ› shows ‹φ → ψ›
103 using assms by (simp add: AOT_sem_imp)
104lemmas CP = "deduction-theorem"
105lemmas "→I" = "deduction-theorem"
106
107AOT_theorem "ded-thm-cor:1": assumes ‹Γ⇩1 ❙⊢ φ → ψ› and ‹Γ⇩2 ❙⊢ ψ → χ› shows ‹Γ⇩1, Γ⇩2 ❙⊢ φ → χ›
108 using "→E" "→I" assms by blast
109AOT_theorem "ded-thm-cor:2": assumes ‹Γ⇩1 ❙⊢ φ → (ψ → χ)› and ‹Γ⇩2 ❙⊢ ψ› shows ‹Γ⇩1, Γ⇩2 ❙⊢ φ → χ›
110 using "→E" "→I" assms by blast
111
112AOT_theorem "ded-thm-cor:3": assumes ‹φ → ψ› and ‹ψ → χ› shows ‹φ → χ›
113 using "→E" "→I" assms by blast
114declare "ded-thm-cor:3"[trans]
115AOT_theorem "ded-thm-cor:4": assumes ‹φ → (ψ → χ)› and ‹ψ› shows ‹φ → χ›
116 using "→E" "→I" assms by blast
117
118lemmas "Hypothetical Syllogism" = "ded-thm-cor:3"
119
120AOT_theorem "useful-tautologies:1": ‹¬¬φ → φ›
121 by (metis "pl:3"[axiom_inst] "→I" "Hypothetical Syllogism")
122AOT_theorem "useful-tautologies:2": ‹φ → ¬¬φ›
123 by (metis "pl:3"[axiom_inst] "→I" "ded-thm-cor:4")
124AOT_theorem "useful-tautologies:3": ‹¬φ → (φ → ψ)›
125 by (meson "ded-thm-cor:4" "pl:3"[axiom_inst] "→I")
126AOT_theorem "useful-tautologies:4": ‹(¬ψ → ¬φ) → (φ → ψ)›
127 by (meson "pl:3"[axiom_inst] "Hypothetical Syllogism" "→I")
128AOT_theorem "useful-tautologies:5": ‹(φ → ψ) → (¬ψ → ¬φ)›
129 by (metis "useful-tautologies:4" "Hypothetical Syllogism" "→I")
130
131AOT_theorem "useful-tautologies:6": ‹(φ → ¬ψ) → (ψ → ¬φ)›
132 by (metis "→I" MP "useful-tautologies:4")
133
134AOT_theorem "useful-tautologies:7": ‹(¬φ → ψ) → (¬ψ → φ)›
135 by (metis "→I" MP "useful-tautologies:3" "useful-tautologies:5")
136
137AOT_theorem "useful-tautologies:8": ‹φ → (¬ψ → ¬(φ → ψ))›
138 by (metis "→I" MP "useful-tautologies:5")
139
140AOT_theorem "useful-tautologies:9": ‹(φ → ψ) → ((¬φ → ψ) → ψ)›
141 by (metis "→I" MP "useful-tautologies:6")
142
143AOT_theorem "useful-tautologies:10": ‹(φ → ¬ψ) → ((φ → ψ) → ¬φ)›
144 by (metis "→I" MP "pl:3"[axiom_inst])
145
146AOT_theorem "dn-i-e:1": assumes ‹φ› shows ‹¬¬φ›
147 using MP "useful-tautologies:2" assms by blast
148lemmas "¬¬I" = "dn-i-e:1"
149AOT_theorem "dn-i-e:2": assumes ‹¬¬φ› shows ‹φ›
150 using MP "useful-tautologies:1" assms by blast
151lemmas "¬¬E" = "dn-i-e:2"
152
153AOT_theorem "modus-tollens:1": assumes ‹φ → ψ› and ‹¬ψ› shows ‹¬φ›
154 using MP "useful-tautologies:5" assms by blast
155AOT_theorem "modus-tollens:2": assumes ‹φ → ¬ψ› and ‹ψ› shows ‹¬φ›
156 using "¬¬I" "modus-tollens:1" assms by blast
157lemmas MT = "modus-tollens:1" "modus-tollens:2"
158
159AOT_theorem "contraposition:1[1]": assumes ‹φ → ψ› shows ‹¬ψ → ¬φ›
160 using "→I" MT(1) assms by blast
161AOT_theorem "contraposition:1[2]": assumes ‹¬ψ → ¬φ› shows ‹φ → ψ›
162 using "→I" "¬¬E" MT(2) assms by blast
163
164AOT_theorem "contraposition:2": assumes ‹φ → ¬ψ› shows ‹ψ → ¬φ›
165 using "→I" MT(2) assms by blast
166
167
168AOT_theorem "reductio-aa:1":
169 assumes ‹¬φ ❙⊢ ¬ψ› and ‹¬φ ❙⊢ ψ› shows ‹φ›
170 using "→I" "¬¬E" MT(2) assms by blast
171AOT_theorem "reductio-aa:2":
172 assumes ‹φ ❙⊢ ¬ψ› and ‹φ ❙⊢ ψ› shows ‹¬φ›
173 using "reductio-aa:1" assms by blast
174lemmas "RAA" = "reductio-aa:1" "reductio-aa:2"
175
176AOT_theorem "exc-mid": ‹φ ∨ ¬φ›
177 using "df-rules-formulas[4]" "if-p-then-p" MP "conventions:2" by blast
178
179AOT_theorem "non-contradiction": ‹¬(φ & ¬φ)›
180 using "df-rules-formulas[3]" MT(2) "useful-tautologies:2" "conventions:1" by blast
181
182AOT_theorem "con-dis-taut:1": ‹(φ & ψ) → φ›
183 by (meson "→I" "df-rules-formulas[3]" MP RAA(1) "conventions:1")
184AOT_theorem "con-dis-taut:2": ‹(φ & ψ) → ψ›
185 by (metis "→I" "df-rules-formulas[3]" MT(2) RAA(2) "¬¬E" "conventions:1")
186lemmas "Conjunction Simplification" = "con-dis-taut:1" "con-dis-taut:2"
187
188AOT_theorem "con-dis-taut:3": ‹φ → (φ ∨ ψ)›
189 by (meson "contraposition:1[2]" "df-rules-formulas[4]" MP "→I" "conventions:2")
190AOT_theorem "con-dis-taut:4": ‹ψ → (φ ∨ ψ)›
191 using "Hypothetical Syllogism" "df-rules-formulas[4]" "pl:1"[axiom_inst] "conventions:2" by blast
192lemmas "Disjunction Addition" = "con-dis-taut:3" "con-dis-taut:4"
193
194AOT_theorem "con-dis-taut:5": ‹φ → (ψ → (φ & ψ))›
195 by (metis "contraposition:2" "Hypothetical Syllogism" "→I" "df-rules-formulas[4]" "conventions:1")
196lemmas Adjunction = "con-dis-taut:5"
197
198AOT_theorem "con-dis-taut:6": ‹(φ & φ) ≡ φ›
199 by (metis Adjunction "→I" "df-rules-formulas[4]" MP "Conjunction Simplification"(1) "conventions:3")
200lemmas "Idempotence of &" = "con-dis-taut:6"
201
202AOT_theorem "con-dis-taut:7": ‹(φ ∨ φ) ≡ φ›
203proof -
204 {
205 AOT_assume ‹φ ∨ φ›
206 AOT_hence ‹¬φ → φ›
207 using "conventions:2"[THEN "df-rules-formulas[3]"] MP by blast
208 AOT_hence ‹φ› using "if-p-then-p" RAA(1) MP by blast
209 }
210 moreover {
211 AOT_assume ‹φ›
212 AOT_hence ‹φ ∨ φ› using "Disjunction Addition"(1) MP by blast
213 }
214 ultimately AOT_show ‹(φ ∨ φ) ≡ φ›
215 using "conventions:3"[THEN "df-rules-formulas[4]"] MP
216 by (metis Adjunction "→I")
217qed
218lemmas "Idempotence of ∨" = "con-dis-taut:7"
219
220
221AOT_theorem "con-dis-i-e:1": assumes ‹φ› and ‹ψ› shows ‹φ & ψ›
222 using Adjunction MP assms by blast
223lemmas "&I" = "con-dis-i-e:1"
224
225AOT_theorem "con-dis-i-e:2:a": assumes ‹φ & ψ› shows ‹φ›
226 using "Conjunction Simplification"(1) MP assms by blast
227AOT_theorem "con-dis-i-e:2:b": assumes ‹φ & ψ› shows ‹ψ›
228 using "Conjunction Simplification"(2) MP assms by blast
229lemmas "&E" = "con-dis-i-e:2:a" "con-dis-i-e:2:b"
230
231AOT_theorem "con-dis-i-e:3:a": assumes ‹φ› shows ‹φ ∨ ψ›
232 using "Disjunction Addition"(1) MP assms by blast
233AOT_theorem "con-dis-i-e:3:b": assumes ‹ψ› shows ‹φ ∨ ψ›
234 using "Disjunction Addition"(2) MP assms by blast
235AOT_theorem "con-dis-i-e:3:c": assumes ‹φ ∨ ψ› and ‹φ → χ› and ‹ψ → Θ› shows ‹χ ∨ Θ›
236 by (metis "con-dis-i-e:3:a" "Disjunction Addition"(2) "df-rules-formulas[3]" MT(1) RAA(1) "conventions:2" assms)
237lemmas "∨I" = "con-dis-i-e:3:a" "con-dis-i-e:3:b" "con-dis-i-e:3:c"
238
239AOT_theorem "con-dis-i-e:4:a": assumes ‹φ ∨ ψ› and ‹φ → χ› and ‹ψ → χ› shows ‹χ›
240 by (metis MP RAA(2) "df-rules-formulas[3]" "conventions:2" assms)
241AOT_theorem "con-dis-i-e:4:b": assumes ‹φ ∨ ψ› and ‹¬φ› shows ‹ψ›
242 using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
243AOT_theorem "con-dis-i-e:4:c": assumes ‹φ ∨ ψ› and ‹¬ψ› shows ‹φ›
244 using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
245lemmas "∨E" = "con-dis-i-e:4:a" "con-dis-i-e:4:b" "con-dis-i-e:4:c"
246
247AOT_theorem "raa-cor:1": assumes ‹¬φ ❙⊢ ψ & ¬ψ› shows ‹φ›
248 using "&E" "∨E"(3) "∨I"(2) RAA(2) assms by blast
249AOT_theorem "raa-cor:2": assumes ‹φ ❙⊢ ψ & ¬ψ› shows ‹¬φ›
250 using "raa-cor:1" assms by blast
251AOT_theorem "raa-cor:3": assumes ‹φ› and ‹¬ψ ❙⊢ ¬φ› shows ‹ψ›
252 using RAA assms by blast
253AOT_theorem "raa-cor:4": assumes ‹¬φ› and ‹¬ψ ❙⊢ φ› shows ‹ψ›
254 using RAA assms by blast
255AOT_theorem "raa-cor:5": assumes ‹φ› and ‹ψ ❙⊢ ¬φ› shows ‹¬ψ›
256 using RAA assms by blast
257AOT_theorem "raa-cor:6": assumes ‹¬φ› and ‹ψ ❙⊢ φ› shows ‹¬ψ›
258 using RAA assms by blast
259
260
261AOT_theorem "oth-class-taut:1:a": ‹(φ → ψ) ≡ ¬(φ & ¬ψ)›
262 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
263 (metis "&E" "&I" "raa-cor:3" "→I" MP)
264AOT_theorem "oth-class-taut:1:b": ‹¬(φ → ψ) ≡ (φ & ¬ψ)›
265 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
266 (metis "&E" "&I" "raa-cor:3" "→I" MP)
267AOT_theorem "oth-class-taut:1:c": ‹(φ → ψ) ≡ (¬φ ∨ ψ)›
268 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
269 (metis "&I" "∨I"(1, 2) "∨E"(3) "→I" MP "raa-cor:1")
270
271AOT_theorem "oth-class-taut:2:a": ‹(φ & ψ) ≡ (ψ & φ)›
272 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
273 (meson "&I" "&E" "→I")
274lemmas "Commutativity of &" = "oth-class-taut:2:a"
275AOT_theorem "oth-class-taut:2:b": ‹(φ & (ψ & χ)) ≡ ((φ & ψ) & χ)›
276 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
277 (metis "&I" "&E" "→I")
278lemmas "Associativity of &" = "oth-class-taut:2:b"
279AOT_theorem "oth-class-taut:2:c": ‹(φ ∨ ψ) ≡ (ψ ∨ φ)›
280 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
281 (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
282lemmas "Commutativity of ∨" = "oth-class-taut:2:c"
283AOT_theorem "oth-class-taut:2:d": ‹(φ ∨ (ψ ∨ χ)) ≡ ((φ ∨ ψ) ∨ χ)›
284 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
285 (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
286lemmas "Associativity of ∨" = "oth-class-taut:2:d"
287AOT_theorem "oth-class-taut:2:e": ‹(φ ≡ ψ) ≡ (ψ ≡ φ)›
288 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I";
289 metis "&I" "df-rules-formulas[4]" "conventions:3" "&E" "Hypothetical Syllogism" "→I" "df-rules-formulas[3]")
290lemmas "Commutativity of ≡" = "oth-class-taut:2:e"
291AOT_theorem "oth-class-taut:2:f": ‹(φ ≡ (ψ ≡ χ)) ≡ ((φ ≡ ψ) ≡ χ)›
292 using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
293 "→I" "→E" "&E" "&I"
294 by metis
295lemmas "Associativity of ≡" = "oth-class-taut:2:f"
296
297AOT_theorem "oth-class-taut:3:a": ‹φ ≡ φ›
298 using "&I" "vdash-properties:6" "if-p-then-p" "df-rules-formulas[4]" "conventions:3" by blast
299AOT_theorem "oth-class-taut:3:b": ‹φ ≡ ¬¬φ›
300 using "&I" "useful-tautologies:1" "useful-tautologies:2" "vdash-properties:6" "df-rules-formulas[4]" "conventions:3" by blast
301AOT_theorem "oth-class-taut:3:c": ‹¬(φ ≡ ¬φ)›
302 by (metis "&E" "→E" RAA "df-rules-formulas[3]" "conventions:3")
303
304AOT_theorem "oth-class-taut:4:a": ‹(φ → ψ) → ((ψ → χ) → (φ → χ))›
305 by (metis "→E" "→I")
306AOT_theorem "oth-class-taut:4:b": ‹(φ ≡ ψ) ≡ (¬φ ≡ ¬ψ)›
307 using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
308 "→I" "→E" "&E" "&I" RAA by metis
309AOT_theorem "oth-class-taut:4:c": ‹(φ ≡ ψ) → ((φ → χ) ≡ (ψ → χ))›
310 using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
311 "→I" "→E" "&E" "&I" by metis
312AOT_theorem "oth-class-taut:4:d": ‹(φ ≡ ψ) → ((χ → φ) ≡ (χ → ψ))›
313 using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
314 "→I" "→E" "&E" "&I" by metis
315AOT_theorem "oth-class-taut:4:e": ‹(φ ≡ ψ) → ((φ & χ) ≡ (ψ & χ))›
316 using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
317 "→I" "→E" "&E" "&I" by metis
318AOT_theorem "oth-class-taut:4:f": ‹(φ ≡ ψ) → ((χ & φ) ≡ (χ & ψ))›
319 using "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
320 "→I" "→E" "&E" "&I" by metis
321AOT_theorem "oth-class-taut:4:g": ‹(φ ≡ ψ) ≡ ((φ & ψ) ∨ (¬φ & ¬ψ))›
322proof(safe intro!: "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"] "&I" "→I"
323 dest!: "conventions:3"[THEN "df-rules-formulas[3]", THEN "→E"])
324 AOT_show ‹φ & ψ ∨ (¬φ & ¬ψ)› if ‹(φ → ψ) & (ψ → φ)›
325 using "&E" "∨I" "→E" "&I" "raa-cor:1" "→I" "∨E" that by metis
326next
327 AOT_show ‹ψ› if ‹φ & ψ ∨ (¬φ & ¬ψ)› and ‹φ›
328 using that "∨E" "&E" "raa-cor:3" by blast
329next
330 AOT_show ‹φ› if ‹φ & ψ ∨ (¬φ & ¬ψ)› and ‹ψ›
331 using that "∨E" "&E" "raa-cor:3" by blast
332qed
333AOT_theorem "oth-class-taut:4:h": ‹¬(φ ≡ ψ) ≡ ((φ & ¬ψ) ∨ (¬φ & ψ))›
334proof (safe intro!: "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"] "&I" "→I")
335 AOT_show ‹φ & ¬ψ ∨ (¬φ & ψ)› if ‹¬(φ ≡ ψ)›
336 by (metis that "&I" "∨I"(1, 2) "→I" MT(1) "df-rules-formulas[4]" "raa-cor:3" "conventions:3")
337next
338 AOT_show ‹¬(φ ≡ ψ)› if ‹φ & ¬ψ ∨ (¬φ & ψ)›
339 by (metis that "&E" "∨E"(2) "→E" "df-rules-formulas[3]" "raa-cor:3" "conventions:3")
340qed
341AOT_theorem "oth-class-taut:5:a": ‹(φ & ψ) ≡ ¬(¬φ ∨ ¬ψ)›
342 using "conventions:3"[THEN "df-rules-formulas[4]"]
343 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
344AOT_theorem "oth-class-taut:5:b": ‹(φ ∨ ψ) ≡ ¬(¬φ & ¬ψ)›
345 using "conventions:3"[THEN "df-rules-formulas[4]"]
346 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
347AOT_theorem "oth-class-taut:5:c": ‹¬(φ & ψ) ≡ (¬φ ∨ ¬ψ)›
348 using "conventions:3"[THEN "df-rules-formulas[4]"]
349 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
350AOT_theorem "oth-class-taut:5:d": ‹¬(φ ∨ ψ) ≡ (¬φ & ¬ψ)›
351 using "conventions:3"[THEN "df-rules-formulas[4]"]
352 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
353
354lemmas DeMorgan = "oth-class-taut:5:c" "oth-class-taut:5:d"
355
356AOT_theorem "oth-class-taut:6:a": ‹(φ & (ψ ∨ χ)) ≡ ((φ & ψ) ∨ (φ & χ))›
357 using "conventions:3"[THEN "df-rules-formulas[4]"]
358 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
359AOT_theorem "oth-class-taut:6:b": ‹(φ ∨ (ψ & χ)) ≡ ((φ ∨ ψ) & (φ ∨ χ))›
360 using "conventions:3"[THEN "df-rules-formulas[4]"]
361 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
362
363AOT_theorem "oth-class-taut:7:a": ‹((φ & ψ) → χ) → (φ → (ψ → χ))›
364 by (metis "&I" "→E" "→I")
365lemmas Exportation = "oth-class-taut:7:a"
366AOT_theorem "oth-class-taut:7:b": ‹(φ → (ψ →χ)) → ((φ & ψ) → χ)›
367 by (metis "&E" "→E" "→I")
368lemmas Importation = "oth-class-taut:7:b"
369
370AOT_theorem "oth-class-taut:8:a": ‹(φ → (ψ → χ)) ≡ (ψ → (φ → χ))›
371 using "conventions:3"[THEN "df-rules-formulas[4]"] "→I" "→E" "&E" "&I" by metis
372lemmas Permutation = "oth-class-taut:8:a"
373AOT_theorem "oth-class-taut:8:b": ‹(φ → ψ) → ((φ → χ) → (φ → (ψ & χ)))›
374 by (metis "&I" "→E" "→I")
375lemmas Composition = "oth-class-taut:8:b"
376AOT_theorem "oth-class-taut:8:c": ‹(φ → χ) → ((ψ → χ) → ((φ ∨ ψ) → χ))›
377 by (metis "∨E"(2) "→E" "→I" RAA(1))
378AOT_theorem "oth-class-taut:8:d": ‹((φ → ψ) & (χ → Θ)) → ((φ & χ) → (ψ & Θ))›
379 by (metis "&E" "&I" "→E" "→I")
380lemmas "Double Composition" = "oth-class-taut:8:d"
381AOT_theorem "oth-class-taut:8:e": ‹((φ & ψ) ≡ (φ & χ)) ≡ (φ → (ψ ≡ χ))›
382 by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
383 "→I" "→E" "&E" "&I")
384AOT_theorem "oth-class-taut:8:f": ‹((φ & ψ) ≡ (χ & ψ)) ≡ (ψ → (φ ≡ χ))›
385 by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
386 "→I" "→E" "&E" "&I")
387AOT_theorem "oth-class-taut:8:g": ‹(ψ ≡ χ) → ((φ ∨ ψ) ≡ (φ ∨ χ))›
388 by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
389 "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
390AOT_theorem "oth-class-taut:8:h": ‹(ψ ≡ χ) → ((ψ ∨ φ) ≡ (χ ∨ φ))›
391 by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
392 "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
393AOT_theorem "oth-class-taut:8:i": ‹(φ ≡ (ψ & χ)) → (ψ → (φ ≡ χ))›
394 by (metis "conventions:3"[THEN "df-rules-formulas[4]"] "conventions:3"[THEN "df-rules-formulas[3]"]
395 "→I" "→E" "&E" "&I")
396
397AOT_theorem "intro-elim:1": assumes ‹φ ∨ ψ› and ‹φ ≡ χ› and ‹ψ ≡ Θ› shows ‹χ ∨ Θ›
398 by (metis assms "∨I"(1, 2) "∨E"(1) "conventions:3"[THEN "df-rules-formulas[3]"] "→I" "→E" "&E"(1))
399
400AOT_theorem "intro-elim:2": assumes ‹φ → ψ› and ‹ψ → φ› shows ‹φ ≡ ψ›
401 by (meson "&I" "conventions:3" "df-rules-formulas[4]" MP assms)
402lemmas "≡I" = "intro-elim:2"
403
404AOT_theorem "intro-elim:3:a": assumes ‹φ ≡ ψ› and ‹φ› shows ‹ψ›
405 by (metis "∨I"(1) "→I" "∨E"(1) "intro-elim:1" assms)
406AOT_theorem "intro-elim:3:b": assumes ‹φ ≡ ψ› and ‹ψ› shows ‹φ›
407 using "intro-elim:3:a" "Commutativity of ≡" assms by blast
408AOT_theorem "intro-elim:3:c": assumes ‹φ ≡ ψ› and ‹¬φ› shows ‹¬ψ›
409 using "intro-elim:3:b" "raa-cor:3" assms by blast
410AOT_theorem "intro-elim:3:d": assumes ‹φ ≡ ψ› and ‹¬ψ› shows ‹¬φ›
411 using "intro-elim:3:a" "raa-cor:3" assms by blast
412AOT_theorem "intro-elim:3:e": assumes ‹φ ≡ ψ› and ‹ψ ≡ χ› shows ‹φ ≡ χ›
413 by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
414declare "intro-elim:3:e"[trans]
415AOT_theorem "intro-elim:3:f": assumes ‹φ ≡ ψ› and ‹φ ≡ χ› shows ‹χ ≡ ψ›
416 by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
417lemmas "≡E" = "intro-elim:3:a" "intro-elim:3:b" "intro-elim:3:c" "intro-elim:3:d" "intro-elim:3:e" "intro-elim:3:f"
418
419declare "Commutativity of ≡"[THEN "≡E"(1), sym]
420
421AOT_theorem "rule-eq-df:1": assumes ‹φ ≡⇩d⇩f ψ› shows ‹φ ≡ ψ›
422 by (simp add: "≡I" "df-rules-formulas[3]" "df-rules-formulas[4]" assms)
423lemmas "≡Df" = "rule-eq-df:1"
424AOT_theorem "rule-eq-df:2": assumes ‹φ ≡⇩d⇩f ψ› and ‹φ› shows ‹ψ›
425 using "≡Df" "≡E"(1) assms by blast
426lemmas "≡⇩d⇩fE" = "rule-eq-df:2"
427AOT_theorem "rule-eq-df:3": assumes ‹φ ≡⇩d⇩f ψ› and ‹ψ› shows ‹φ›
428 using "≡Df" "≡E"(2) assms by blast
429lemmas "≡⇩d⇩fI" = "rule-eq-df:3"
430
431AOT_theorem "df-simplify:1": assumes ‹φ ≡ (ψ & χ)› and ‹ψ› shows ‹φ ≡ χ›
432 by (metis "&E"(2) "&I" "≡E"(1, 2) "≡I" "→I" assms)
433
434AOT_theorem "df-simplify:2": assumes ‹φ ≡ (ψ & χ)› and ‹χ› shows ‹φ ≡ ψ›
435 by (metis "&E"(1) "&I" "≡E"(1, 2) "≡I" "→I" assms)
436lemmas "≡S" = "df-simplify:1" "df-simplify:2"
437
438AOT_theorem "rule-ui:1": assumes ‹∀α φ{α}› and ‹τ↓› shows ‹φ{τ}›
439 using "→E" "cqt:1"[axiom_inst] assms by blast
440AOT_theorem "rule-ui:2[const_var]": assumes ‹∀α φ{α}› shows ‹φ{β}›
441 by (simp add: "rule-ui:1" "cqt:2[const_var]"[axiom_inst] assms)
442
443AOT_theorem "rule-ui:2[lambda]":
444 assumes ‹∀F φ{F}› and ‹INSTANCE_OF_CQT_2(ψ)›
445 shows ‹φ{[λν⇩1...ν⇩n ψ{ν⇩1...ν⇩n}]}›
446 by (simp add: "rule-ui:1" "cqt:2[lambda]"[axiom_inst] assms)
447AOT_theorem "rule-ui:3": assumes ‹∀α φ{α}› shows ‹φ{α}›
448 by (simp add: "rule-ui:2[const_var]" assms)
449lemmas "∀E" = "rule-ui:1" "rule-ui:2[const_var]" "rule-ui:2[lambda]" "rule-ui:3"
450
451AOT_theorem "cqt-orig:1[const_var]": ‹∀α φ{α} → φ{β}› by (simp add: "∀E"(2) "→I")
452AOT_theorem "cqt-orig:1[lambda]":
453 assumes ‹INSTANCE_OF_CQT_2(ψ)›
454 shows ‹∀F φ{F} → φ{[λν⇩1...ν⇩n ψ{ν⇩1...ν⇩n}]}›
455 by (simp add: "∀E"(3) "→I" assms)
456AOT_theorem "cqt-orig:2": ‹∀α (φ → ψ{α}) → (φ → ∀α ψ{α})›
457 by (metis "→I" GEN "vdash-properties:6" "∀E"(4))
458AOT_theorem "cqt-orig:3": ‹∀α φ{α} → φ{α}› using "cqt-orig:1[const_var]" .
459
460
461AOT_theorem universal: assumes ‹for arbitrary β: φ{β}› shows ‹∀α φ{α}›
462 using GEN assms .
463lemmas "∀I" = universal
464
465
466ML‹
467fun get_instantiated_allI ctxt varname thm = let
468val trm = Thm.concl_of thm
469val trm = case trm of (@{const Trueprop} $ (@{const AOT_model_valid_in} $ _ $ x)) => x
470 | _ => raise Term.TERM ("Expected simple theorem.", [trm])
471fun extractVars (Const (\<^const_name>‹AOT_term_of_var›, _) $ Var v) =
472 (if fst (fst v) = fst varname then [Var v] else [])
473 | extractVars (t1 $ t2) = extractVars t1 @ extractVars t2
474 | extractVars (Abs (_, _, t)) = extractVars t
475 | extractVars _ = []
476val vars = extractVars trm
477val vars = fold Term.add_vars vars []
478val var = hd vars
479val trmty = case (snd var) of (Type (\<^type_name>‹AOT_var›, [t])) => (t)
480 | _ => raise Term.TYPE ("Expected variable type.", [snd var], [Var var])
481val trm = Abs (Term.string_of_vname (fst var), trmty, Term.abstract_over (
482 Const (\<^const_name>‹AOT_term_of_var›, Type ("fun", [snd var, trmty]))
483 $ Var var, trm))
484val trm = Thm.cterm_of (Context.proof_of ctxt) trm
485val ty = hd (Term.add_tvars (Thm.prop_of @{thm "∀I"}) [])
486val typ = Thm.ctyp_of (Context.proof_of ctxt) trmty
487val allthm = Drule.instantiate_normalize ([(ty, typ)],[]) @{thm "∀I"}
488val phi = hd (Term.add_vars (Thm.prop_of allthm) [])
489val allthm = Drule.instantiate_normalize ([],[(phi,trm)]) allthm
490in
491allthm
492end
493›
494
495attribute_setup "∀I" =
496 ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
497 (fn ctxt => fn thm => fold (fn arg => fn thm => thm RS get_instantiated_allI ctxt arg thm) args thm))›
498 "Quantify over a variable in a theorem using GEN."
499
500attribute_setup "unvarify" =
501 ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
502 (fn ctxt => fn thm =>
503 let
504 val thm = fold (fn arg => fn thm => thm RS get_instantiated_allI ctxt arg thm) args thm
505 val thm = fold (fn _ => fn thm => thm RS @{thm "∀E"(1)}) args thm
506 in
507 thm
508 end))›
509 "Generalize a statement about variables to a statement about denoting terms."
510
511
512
513AOT_theorem "cqt-basic:1": ‹∀α∀β φ{α,β} ≡ ∀β∀α φ{α,β}›
514 by (metis "≡I" "∀E"(2) "∀I" "→I")
515
516AOT_theorem "cqt-basic:2": ‹∀α(φ{α} ≡ ψ{α}) ≡ (∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α}))›
517proof (rule "≡I"; rule "→I")
518 AOT_assume ‹∀α(φ{α} ≡ ψ{α})›
519 AOT_hence ‹φ{α} ≡ ψ{α}› for α using "∀E"(2) by blast
520 AOT_hence ‹φ{α} → ψ{α}› and ‹ψ{α} → φ{α}› for α
521 using "≡E"(1,2) "→I" by blast+
522 AOT_thus ‹∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α})›
523 by (auto intro: "&I" "∀I")
524next
525 AOT_assume ‹∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α})›
526 AOT_hence ‹φ{α} → ψ{α}› and ‹ψ{α} → φ{α}› for α
527 using "∀E"(2) "&E" by blast+
528 AOT_hence ‹φ{α} ≡ ψ{α}› for α
529 using "≡I" by blast
530 AOT_thus ‹∀α(φ{α} ≡ ψ{α})› by (auto intro: "∀I")
531qed
532
533AOT_theorem "cqt-basic:3": ‹∀α(φ{α} ≡ ψ{α}) → (∀α φ{α} ≡ ∀α ψ{α})›
534proof(rule "→I")
535 AOT_assume ‹∀α(φ{α} ≡ ψ{α})›
536 AOT_hence 1: ‹φ{α} ≡ ψ{α}› for α using "∀E"(2) by blast
537 {
538 AOT_assume ‹∀α φ{α}›
539 AOT_hence ‹∀α ψ{α}› using 1 "∀I" "∀E"(4) "≡E" by metis
540 }
541 moreover {
542 AOT_assume ‹∀α ψ{α}›
543 AOT_hence ‹∀α φ{α}› using 1 "∀I" "∀E"(4) "≡E" by metis
544 }
545 ultimately AOT_show ‹∀α φ{α} ≡ ∀α ψ{α}›
546 using "≡I" "→I" by auto
547qed
548
549AOT_theorem "cqt-basic:4": ‹∀α(φ{α} & ψ{α}) → (∀α φ{α} & ∀α ψ{α})›
550proof(rule "→I")
551 AOT_assume 0: ‹∀α(φ{α} & ψ{α})›
552 AOT_have ‹φ{α}› and ‹ψ{α}› for α using "∀E"(2) 0 "&E" by blast+
553 AOT_thus ‹∀α φ{α} & ∀α ψ{α}›
554 by (auto intro: "∀I" "&I")
555qed
556
557AOT_theorem "cqt-basic:5": ‹(∀α⇩1...∀α⇩n(φ{α⇩1...α⇩n})) → φ{α⇩1...α⇩n}›
558 using "cqt-orig:3" by blast
559
560AOT_theorem "cqt-basic:6": ‹∀α∀α φ{α} ≡ ∀α φ{α}›
561 by (meson "≡I" "→I" GEN "cqt-orig:1[const_var]")
562
563AOT_theorem "cqt-basic:7": ‹(φ → ∀α ψ{α}) ≡ ∀α(φ → ψ{α})›
564 by (metis "→I" "vdash-properties:6" "rule-ui:3" "≡I" GEN)
565
566AOT_theorem "cqt-basic:8": ‹(∀α φ{α} ∨ ∀α ψ{α}) → ∀α (φ{α} ∨ ψ{α})›
567 by (simp add: "∨I"(3) "→I" GEN "cqt-orig:1[const_var]")
568
569AOT_theorem "cqt-basic:9": ‹(∀α (φ{α} → ψ{α}) & ∀α (ψ{α} → χ{α})) → ∀α(φ{α} → χ{α})›
570proof -
571 {
572 AOT_assume ‹∀α (φ{α} → ψ{α})›
573 moreover AOT_assume ‹∀α (ψ{α} → χ{α})›
574 ultimately AOT_have ‹φ{α} → ψ{α}› and ‹ψ{α} → χ{α}› for α using "∀E" by blast+
575 AOT_hence ‹φ{α} → χ{α}› for α by (metis "→E" "→I")
576 AOT_hence ‹∀α(φ{α} → χ{α})› using "∀I" by fast
577 }
578 thus ?thesis using "&I" "→I" "&E" by meson
579qed
580
581AOT_theorem "cqt-basic:10": ‹(∀α(φ{α} ≡ ψ{α}) & ∀α(ψ{α} ≡ χ{α})) → ∀α (φ{α} ≡ χ{α})›
582proof(rule "→I"; rule "∀I")
583 fix β
584 AOT_assume ‹∀α(φ{α} ≡ ψ{α}) & ∀α(ψ{α} ≡ χ{α})›
585 AOT_hence ‹φ{β} ≡ ψ{β}› and ‹ψ{β} ≡ χ{β}› using "&E" "∀E" by blast+
586 AOT_thus ‹φ{β} ≡ χ{β}› using "≡I" "≡E" by blast
587qed
588
589AOT_theorem "cqt-basic:11": ‹∀α(φ{α} ≡ ψ{α}) ≡ ∀α (ψ{α} ≡ φ{α})›
590proof (rule "≡I"; rule "→I")
591 AOT_assume 0: ‹∀α(φ{α} ≡ ψ{α})›
592 {
593 fix α
594 AOT_have ‹φ{α} ≡ ψ{α}› using 0 "∀E" by blast
595 AOT_hence ‹ψ{α} ≡ φ{α}› using "≡I" "≡E" "→I" "→E" by metis
596 }
597 AOT_thus ‹∀α(ψ{α} ≡ φ{α})› using "∀I" by fast
598next
599 AOT_assume 0: ‹∀α(ψ{α} ≡ φ{α})›
600 {
601 fix α
602 AOT_have ‹ψ{α} ≡ φ{α}› using 0 "∀E" by blast
603 AOT_hence ‹φ{α} ≡ ψ{α}› using "≡I" "≡E" "→I" "→E" by metis
604 }
605 AOT_thus ‹∀α(φ{α} ≡ ψ{α})› using "∀I" by fast
606qed
607
608AOT_theorem "cqt-basic:12": ‹∀α φ{α} → ∀α (ψ{α} → φ{α})›
609 by (simp add: "∀E"(2) "→I" GEN)
610
611AOT_theorem "cqt-basic:13": ‹∀α φ{α} ≡ ∀β φ{β}›
612 using "≡I" "→I" by blast
613
614AOT_theorem "cqt-basic:14": ‹(∀α⇩1...∀α⇩n (φ{α⇩1...α⇩n} → ψ{α⇩1...α⇩n})) → ((∀α⇩1...∀α⇩n φ{α⇩1...α⇩n}) → (∀α⇩1...∀α⇩n ψ{α⇩1...α⇩n}))›
615 using "cqt:3"[axiom_inst] by auto
616
617AOT_theorem "cqt-basic:15": ‹(∀α⇩1...∀α⇩n (φ → ψ{α⇩1...α⇩n})) → (φ → (∀α⇩1...∀α⇩n ψ{α⇩1...α⇩n}))›
618 using "cqt-orig:2" by auto
619
620
621AOT_theorem "universal-cor": assumes ‹for arbitrary β: φ{β}› shows ‹∀α φ{α}›
622 using GEN assms .
623
624AOT_theorem "existential:1": assumes ‹φ{τ}› and ‹τ↓› shows ‹∃α φ{α}›
625proof(rule "raa-cor:1")
626 AOT_assume ‹¬∃α φ{α}›
627 AOT_hence ‹∀α ¬φ{α}›
628 using "≡⇩d⇩fI" "conventions:4" RAA "&I" by blast
629 AOT_hence ‹¬φ{τ}› using assms(2) "∀E"(1) "→E" by blast
630 AOT_thus ‹φ{τ} & ¬φ{τ}› using assms(1) "&I" by blast
631qed
632
633AOT_theorem "existential:2[const_var]": assumes ‹φ{β}› shows ‹∃α φ{α}›
634 using "existential:1" "cqt:2[const_var]"[axiom_inst] assms by blast
635
636AOT_theorem "existential:2[lambda]":
637 assumes ‹φ{[λν⇩1...ν⇩n ψ{ν⇩1...ν⇩n}]}› and ‹INSTANCE_OF_CQT_2(ψ)›
638 shows ‹∃α φ{α}›
639 using "existential:1" "cqt:2[lambda]"[axiom_inst] assms by blast
640lemmas "∃I" = "existential:1" "existential:2[const_var]" "existential:2[lambda]"
641
642AOT_theorem "instantiation":
643 assumes ‹for arbitrary β: φ{β} ❙⊢ ψ› and ‹∃α φ{α}›
644 shows ‹ψ›
645 by (metis (no_types, lifting) "≡⇩d⇩fE" GEN "raa-cor:3" "conventions:4" assms)
646lemmas "∃E" = "instantiation"
647
648AOT_theorem "cqt-further:1": ‹∀α φ{α} → ∃α φ{α}›
649 using "∀E"(4) "∃I"(2) "→I" by metis
650
651AOT_theorem "cqt-further:2": ‹¬∀α φ{α} → ∃α ¬φ{α}›
652 using "∀I" "∃I"(2) "→I" RAA by metis
653
654AOT_theorem "cqt-further:3": ‹∀α φ{α} ≡ ¬∃α ¬φ{α}›
655 using "∀E"(4) "∃E" "→I" RAA
656 by (metis "cqt-further:2" "≡I" "modus-tollens:1")
657
658AOT_theorem "cqt-further:4": ‹¬∃α φ{α} → ∀α ¬φ{α}›
659 using "∀I" "∃I"(2)"→I" RAA by metis
660
661AOT_theorem "cqt-further:5": ‹∃α (φ{α} & ψ{α}) → (∃α φ{α} & ∃α ψ{α})›
662 by (metis (no_types, lifting) "&E" "&I" "∃E" "∃I"(2) "→I")
663
664AOT_theorem "cqt-further:6": ‹∃α (φ{α} ∨ ψ{α}) → (∃α φ{α} ∨ ∃α ψ{α})›
665 by (metis (mono_tags, lifting) "∃E" "∃I"(2) "∨E"(3) "∨I"(1, 2) "→I" RAA(2))
666
667AOT_theorem "cqt-further:7": ‹∃α φ{α} ≡ ∃β φ{β}›
668 by (simp add: "oth-class-taut:3:a")
669
670AOT_theorem "cqt-further:8": ‹(∀α φ{α} & ∀α ψ{α}) → ∀α (φ{α} ≡ ψ{α})›
671 by (metis (mono_tags, lifting) "&E" "≡I" "∀E"(2) "→I" GEN)
672
673AOT_theorem "cqt-further:9": ‹(¬∃α φ{α} & ¬∃α ψ{α}) → ∀α (φ{α} ≡ ψ{α})›
674 by (metis (mono_tags, lifting) "&E" "≡I" "∃I"(2) "→I" GEN "raa-cor:4")
675
676AOT_theorem "cqt-further:10": ‹(∃α φ{α} & ¬∃α ψ{α}) → ¬∀α (φ{α} ≡ ψ{α})›
677proof(rule "→I"; rule "raa-cor:2")
678 AOT_assume 0: ‹∃α φ{α} & ¬∃α ψ{α}›
679 then AOT_obtain α where ‹φ{α}› using "∃E" "&E"(1) by metis
680 moreover AOT_assume ‹∀α (φ{α} ≡ ψ{α})›
681 ultimately AOT_have ‹ψ{α}› using "∀E"(4) "≡E"(1) by blast
682 AOT_hence ‹∃α ψ{α}› using "∃I" by blast
683 AOT_thus ‹∃α ψ{α} & ¬∃α ψ{α}› using 0 "&E"(2) "&I" by blast
684qed
685
686AOT_theorem "cqt-further:11": ‹∃α∃β φ{α,β} ≡ ∃β∃α φ{α,β}›
687 using "≡I" "→I" "∃I"(2) "∃E" by metis
688
689AOT_theorem "log-prop-prop:1": ‹[λ φ]↓›
690 using "cqt:2[lambda0]"[axiom_inst] by auto
691
692AOT_theorem "log-prop-prop:2": ‹φ↓›
693 by (rule "≡⇩d⇩fI"[OF "existence:3"]) "cqt:2[lambda]"
694
695AOT_theorem "exist-nec": ‹τ↓ → □τ↓›
696proof -
697 AOT_have ‹∀β □β↓›
698 by (simp add: GEN RN "cqt:2[const_var]"[axiom_inst])
699 AOT_thus ‹τ↓ → □τ↓›
700 using "cqt:1"[axiom_inst] "→E" by blast
701qed
702
703
704class AOT_Term_id = AOT_Term +
705 assumes "t=t-proper:1"[AOT]: ‹[v ⊨ τ = τ' → τ↓]›
706 and "t=t-proper:2"[AOT]: ‹[v ⊨ τ = τ' → τ'↓]›
707
708instance κ :: AOT_Term_id
709proof
710 AOT_modally_strict {
711 AOT_show ‹κ = κ' → κ↓› for κ κ'
712 proof(rule "→I")
713 AOT_assume ‹κ = κ'›
714 AOT_hence ‹O!κ ∨ A!κ›
715 by (rule "∨I"(3)[OF "≡⇩d⇩fE"[OF "identity:1"]])
716 (meson "→I" "∨I"(1) "&E"(1))+
717 AOT_thus ‹κ↓›
718 by (rule "∨E"(1))
719 (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
720 qed
721 }
722next
723 AOT_modally_strict {
724 AOT_show ‹κ = κ' → κ'↓› for κ κ'
725 proof(rule "→I")
726 AOT_assume ‹κ = κ'›
727 AOT_hence ‹O!κ' ∨ A!κ'›
728 by (rule "∨I"(3)[OF "≡⇩d⇩fE"[OF "identity:1"]])
729 (meson "→I" "∨I" "&E")+
730 AOT_thus ‹κ'↓›
731 by (rule "∨E"(1))
732 (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
733 qed
734 }
735qed
736
737instance rel :: (AOT_κs) AOT_Term_id
738proof
739 AOT_modally_strict {
740 AOT_show ‹Π = Π' → Π↓› for Π Π' :: ‹<'a>›
741 proof(rule "→I")
742 AOT_assume ‹Π = Π'›
743 AOT_thus ‹Π↓› using "≡⇩d⇩fE"[OF "identity:3"[of Π Π']] "&E" by blast
744 qed
745 }
746next
747 AOT_modally_strict {
748 AOT_show ‹Π = Π' → Π'↓› for Π Π' :: ‹<'a>›
749 proof(rule "→I")
750 AOT_assume ‹Π = Π'›
751 AOT_thus ‹Π'↓› using "≡⇩d⇩fE"[OF "identity:3"[of Π Π']] "&E" by blast
752 qed
753 }
754qed
755
756instance 𝗈 :: AOT_Term_id
757proof
758 AOT_modally_strict {
759 fix φ ψ
760 AOT_show ‹φ = ψ → φ↓›
761 proof(rule "→I")
762 AOT_assume ‹φ = ψ›
763 AOT_thus ‹φ↓› using "≡⇩d⇩fE"[OF "identity:4"[of φ ψ]] "&E" by blast
764 qed
765 }
766next
767 AOT_modally_strict {
768 fix φ ψ
769 AOT_show ‹φ = ψ → ψ↓›
770 proof(rule "→I")
771 AOT_assume ‹φ = ψ›
772 AOT_thus ‹ψ↓› using "≡⇩d⇩fE"[OF "identity:4"[of φ ψ]] "&E" by blast
773 qed
774 }
775qed
776
777instance prod :: (AOT_Term_id, AOT_Term_id) AOT_Term_id
778proof
779 AOT_modally_strict {
780 fix τ τ' :: ‹'a×'b›
781 AOT_show ‹τ = τ' → τ↓›
782 proof (induct τ; induct τ'; rule "→I")
783 fix τ⇩1 τ⇩1' :: 'a and τ⇩2 τ⇩2' :: 'b
784 AOT_assume ‹«(τ⇩1, τ⇩2)» = «(τ⇩1', τ⇩2')»›
785 AOT_hence ‹(τ⇩1 = τ⇩1') & (τ⇩2 = τ⇩2')› by (metis "≡⇩d⇩fE" tuple_identity_1)
786 AOT_hence ‹τ⇩1↓› and ‹τ⇩2↓› using "t=t-proper:1" "&E" "vdash-properties:10" by blast+
787 AOT_thus ‹«(τ⇩1, τ⇩2)»↓› by (metis "≡⇩d⇩fI" "&I" tuple_denotes)
788 qed
789 }
790next
791 AOT_modally_strict {
792 fix τ τ' :: ‹'a×'b›
793 AOT_show ‹τ = τ' → τ'↓›
794 proof (induct τ; induct τ'; rule "→I")
795 fix τ⇩1 τ⇩1' :: 'a and τ⇩2 τ⇩2' :: 'b
796 AOT_assume ‹«(τ⇩1, τ⇩2)» = «(τ⇩1', τ⇩2')»›
797 AOT_hence ‹(τ⇩1 = τ⇩1') & (τ⇩2 = τ⇩2')› by (metis "≡⇩d⇩fE" tuple_identity_1)
798 AOT_hence ‹τ⇩1'↓› and ‹τ⇩2'↓› using "t=t-proper:2" "&E" "vdash-properties:10" by blast+
799 AOT_thus ‹«(τ⇩1', τ⇩2')»↓› by (metis "≡⇩d⇩fI" "&I" tuple_denotes)
800 qed
801 }
802qed
803
804
805AOT_register_type_constraints
806 Term: ‹_::AOT_Term_id› ‹_::AOT_Term_id›
807AOT_register_type_constraints
808 Individual: ‹κ› ‹_::{AOT_κs, AOT_Term_id}›
809AOT_register_type_constraints
810 Relation: ‹<_::{AOT_κs, AOT_Term_id}>›
811
812AOT_theorem "id-rel-nec-equiv:1": ‹Π = Π' → □∀x⇩1...∀x⇩n ([Π]x⇩1...x⇩n ≡ [Π']x⇩1...x⇩n)›
813proof(rule "→I")
814 AOT_assume assumption: ‹Π = Π'›
815 AOT_hence ‹Π↓› and ‹Π'↓›
816 using "t=t-proper:1" "t=t-proper:2" MP by blast+
817 moreover AOT_have ‹∀F∀G (F = G → ((□∀x⇩1...∀x⇩n ([F]x⇩1...x⇩n ≡ [F]x⇩1...x⇩n)) → □∀x⇩1...∀x⇩n ([F]x⇩1...x⇩n ≡ [G]x⇩1...x⇩n)))›
818 apply (rule GEN)+ using "l-identity"[axiom_inst] by force
819 ultimately AOT_have ‹Π = Π' → ((□∀x⇩1...∀x⇩n ([Π]x⇩1...x⇩n ≡ [Π]x⇩1...x⇩n)) → □∀x⇩1...∀x⇩n ([Π]x⇩1...x⇩n ≡ [Π']x⇩1...x⇩n))›
820 using "∀E"(1) by blast
821 AOT_hence ‹(□∀x⇩1...∀x⇩n ([Π]x⇩1...x⇩n ≡ [Π]x⇩1...x⇩n)) → □∀x⇩1...∀x⇩n ([Π]x⇩1...x⇩n ≡ [Π']x⇩1...x⇩n)›
822 using assumption "→E" by blast
823 moreover AOT_have ‹□∀x⇩1...∀x⇩n ([Π]x⇩1...x⇩n ≡ [Π]x⇩1...x⇩n)›
824 by (simp add: RN "oth-class-taut:3:a" "universal-cor")
825 ultimately AOT_show ‹□∀x⇩1...∀x⇩n ([Π]x⇩1...x⇩n ≡ [Π']x⇩1...x⇩n)›
826 using "→E" by blast
827qed
828
829AOT_theorem "id-rel-nec-equiv:2": ‹φ = ψ → □(φ ≡ ψ)›
830proof(rule "→I")
831 AOT_assume assumption: ‹φ = ψ›
832 AOT_hence ‹φ↓› and ‹ψ↓›
833 using "t=t-proper:1" "t=t-proper:2" MP by blast+
834 moreover AOT_have ‹∀p∀q (p = q → ((□(p ≡ p) → □(p ≡ q))))›
835 apply (rule GEN)+ using "l-identity"[axiom_inst] by force
836 ultimately AOT_have ‹φ = ψ → (□(φ ≡ φ) → □(φ ≡ ψ))›
837 using "∀E"(1) by blast
838 AOT_hence ‹□(φ ≡ φ) → □(φ ≡ ψ)›
839 using assumption "→E" by blast
840 moreover AOT_have ‹□(φ ≡ φ)›
841 by (simp add: RN "oth-class-taut:3:a" "universal-cor")
842 ultimately AOT_show ‹□(φ ≡ ψ)›
843 using "→E" by blast
844qed
845
846AOT_theorem "rule=E": assumes ‹φ{τ}› and ‹τ = σ› shows ‹φ{σ}›
847proof -
848 AOT_have ‹τ↓› and ‹σ↓› using assms(2) "t=t-proper:1" "t=t-proper:2" "→E" by blast+
849 moreover AOT_have ‹∀α∀β(α = β → (φ{α} → φ{β}))›
850 apply (rule GEN)+ using "l-identity"[axiom_inst] by blast
851 ultimately AOT_have ‹τ = σ → (φ{τ} → φ{σ})›
852 using "∀E"(1) by blast
853 AOT_thus ‹φ{σ}› using assms "→E" by blast
854qed
855
856AOT_theorem "propositions-lemma:1": ‹[λ φ] = φ›
857proof -
858 AOT_have ‹φ↓› by (simp add: "log-prop-prop:2")
859 moreover AOT_have ‹∀p [λ p] = p› using "lambda-predicates:3[zero]"[axiom_inst] "∀I" by fast
860 ultimately AOT_show ‹[λ φ] = φ›
861 using "∀E" by blast
862qed
863
864AOT_theorem "propositions-lemma:2": ‹[λ φ] ≡ φ›
865proof -
866 AOT_have ‹[λ φ] ≡ [λ φ]› by (simp add: "oth-class-taut:3:a")
867 AOT_thus ‹[λ φ] ≡ φ› using "propositions-lemma:1" "rule=E" by blast
868qed
869
870
871
872AOT_theorem "propositions-lemma:6": ‹(φ ≡ ψ) ≡ ([λ φ] ≡ [λ ψ])›
873 by (metis "≡E"(1) "≡E"(5) "Associativity of ≡" "propositions-lemma:2")
874
875
876
877AOT_theorem "oa-exist:1": ‹O!↓›
878proof -
879 AOT_have ‹[λx ◇[E!]x]↓› by "cqt:2[lambda]"
880 AOT_hence 1: ‹O! = [λx ◇[E!]x]› using "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1)] "→E" by blast
881 AOT_show ‹O!↓› using "t=t-proper:1"[THEN "→E", OF 1] by simp
882qed
883
884AOT_theorem "oa-exist:2": ‹A!↓›
885proof -
886 AOT_have ‹[λx ¬◇[E!]x]↓› by "cqt:2[lambda]"
887 AOT_hence 1: ‹A! = [λx ¬◇[E!]x]› using "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1)] "→E" by blast
888 AOT_show ‹A!↓› using "t=t-proper:1"[THEN "→E", OF 1] by simp
889qed
890
891AOT_theorem "oa-exist:3": ‹O!x ∨ A!x›
892proof(rule "raa-cor:1")
893 AOT_assume ‹¬(O!x ∨ A!x)›
894 AOT_hence A: ‹¬O!x› and B: ‹¬A!x›
895 using "Disjunction Addition"(1) "modus-tollens:1" "∨I"(2) "raa-cor:5" by blast+
896 AOT_have C: ‹O! = [λx ◇[E!]x]›
897 by (rule "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1), THEN "→E"]) "cqt:2[lambda]"
898 AOT_have D: ‹A! = [λx ¬◇[E!]x]›
899 by (rule "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1), THEN "→E"]) "cqt:2[lambda]"
900 AOT_have E: ‹¬[λx ◇[E!]x]x›
901 using A C "rule=E" by fast
902 AOT_have F: ‹¬[λx ¬◇[E!]x]x›
903 using B D "rule=E" by fast
904 AOT_have G: ‹[λx ◇[E!]x]x ≡ ◇[E!]x›
905 by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
906 AOT_have H: ‹[λx ¬◇[E!]x]x ≡ ¬◇[E!]x›
907 by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
908 AOT_show ‹¬◇[E!]x & ¬¬◇[E!]x› using G E "≡E" H F "≡E" "&I" by metis
909qed
910
911AOT_theorem "p-identity-thm2:1": ‹F = G ≡ □∀x(x[F] ≡ x[G])›
912proof -
913 AOT_have ‹F = G ≡ F↓ & G↓ & □∀x(x[F] ≡ x[G])›
914 using "identity:2" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
915 moreover AOT_have ‹F↓› and ‹G↓›
916 by (auto simp: "cqt:2[const_var]"[axiom_inst])
917 ultimately AOT_show ‹F = G ≡ □∀x(x[F] ≡ x[G])›
918 using "≡S"(1) "&I" by blast
919qed
920
921AOT_theorem "p-identity-thm2:2[2]": ‹F = G ≡ ∀y⇩1([λx [F]xy⇩1] = [λx [G]xy⇩1] & [λx [F]y⇩1x] = [λx [G]y⇩1x])›
922proof -
923 AOT_have ‹F = G ≡ F↓ & G↓ & ∀y⇩1([λx [F]xy⇩1] = [λx [G]xy⇩1] & [λx [F]y⇩1x] = [λx [G]y⇩1x])›
924 using "identity:3[2]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
925 moreover AOT_have ‹F↓› and ‹G↓›
926 by (auto simp: "cqt:2[const_var]"[axiom_inst])
927 ultimately show ?thesis
928 using "≡S"(1) "&I" by blast
929qed
930
931AOT_theorem "p-identity-thm2:2[3]": ‹F = G ≡ ∀y⇩1∀y⇩2([λx [F]xy⇩1y⇩2] = [λx [G]xy⇩1y⇩2] & [λx [F]y⇩1xy⇩2] = [λx [G]y⇩1xy⇩2] & [λx [F]y⇩1y⇩2x] = [λx [G]y⇩1y⇩2x])›
932proof -
933 AOT_have ‹F = G ≡ F↓ & G↓ & ∀y⇩1∀y⇩2([λx [F]xy⇩1y⇩2] = [λx [G]xy⇩1y⇩2] & [λx [F]y⇩1xy⇩2] = [λx [G]y⇩1xy⇩2] & [λx [F]y⇩1y⇩2x] = [λx [G]y⇩1y⇩2x])›
934 using "identity:3[3]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
935 moreover AOT_have ‹F↓› and ‹G↓›
936 by (auto simp: "cqt:2[const_var]"[axiom_inst])
937 ultimately show ?thesis
938 using "≡S"(1) "&I" by blast
939qed
940
941AOT_theorem "p-identity-thm2:2[4]": ‹F = G ≡ ∀y⇩1∀y⇩2∀y⇩3([λx [F]xy⇩1y⇩2y⇩3] = [λx [G]xy⇩1y⇩2y⇩3] & [λx [F]y⇩1xy⇩2y⇩3] = [λx [G]y⇩1xy⇩2y⇩3] & [λx [F]y⇩1y⇩2xy⇩3] = [λx [G]y⇩1y⇩2xy⇩3] & [λx [F]y⇩1y⇩2y⇩3x] = [λx [G]y⇩1y⇩2y⇩3x])›
942proof -
943 AOT_have ‹F = G ≡ F↓ & G↓ & ∀y⇩1∀y⇩2∀y⇩3([λx [F]xy⇩1y⇩2y⇩3] = [λx [G]xy⇩1y⇩2y⇩3] & [λx [F]y⇩1xy⇩2y⇩3] = [λx [G]y⇩1xy⇩2y⇩3] & [λx [F]y⇩1y⇩2xy⇩3] = [λx [G]y⇩1y⇩2xy⇩3] & [λx [F]y⇩1y⇩2y⇩3x] = [λx [G]y⇩1y⇩2y⇩3x])›
944 using "identity:3[4]" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
945 moreover AOT_have ‹F↓› and ‹G↓›
946 by (auto simp: "cqt:2[const_var]"[axiom_inst])
947 ultimately show ?thesis
948 using "≡S"(1) "&I" by blast
949qed
950
951AOT_theorem "p-identity-thm2:2":
952 ‹F = G ≡ ∀x⇩1...∀x⇩n «AOT_sem_proj_id x⇩1x⇩n (λ τ . «[F]τ») (λ τ . «[G]τ»)»›
953proof -
954 AOT_have ‹F = G ≡ F↓ & G↓ & ∀x⇩1...∀x⇩n «AOT_sem_proj_id x⇩1x⇩n (λ τ . «[F]τ») (λ τ . «[G]τ»)»›
955 using "identity:3" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
956 moreover AOT_have ‹F↓› and ‹G↓›
957 by (auto simp: "cqt:2[const_var]"[axiom_inst])
958 ultimately show ?thesis
959 using "≡S"(1) "&I" by blast
960qed
961
962AOT_theorem "p-identity-thm2:3":
963 ‹p = q ≡ [λx p] = [λx q]›
964proof -
965 AOT_have ‹p = q ≡ p↓ & q↓ & [λx p] = [λx q]›
966 using "identity:4" "df-rules-formulas[3]" "df-rules-formulas[4]" "→E" "&E" "≡I" "→I" by blast
967 moreover AOT_have ‹p↓› and ‹q↓›
968 by (auto simp: "cqt:2[const_var]"[axiom_inst])
969 ultimately show ?thesis
970 using "≡S"(1) "&I" by blast
971qed
972
973class AOT_Term_id_2 = AOT_Term_id + assumes "id-eq:1": ‹[v ⊨ α = α]›
974
975instance κ :: AOT_Term_id_2
976proof
977 AOT_modally_strict {
978 fix x
979 {
980 AOT_assume ‹O!x›
981 moreover AOT_have ‹□∀F([F]x ≡ [F]x)›
982 using RN GEN "oth-class-taut:3:a" by fast
983 ultimately AOT_have ‹O!x & O!x & □∀F([F]x ≡ [F]x)› using "&I" by simp
984 }
985 moreover {
986 AOT_assume ‹A!x›
987 moreover AOT_have ‹□∀F(x[F] ≡ x[F])›
988 using RN GEN "oth-class-taut:3:a" by fast
989 ultimately AOT_have ‹A!x & A!x & □∀F(x[F] ≡ x[F])› using "&I" by simp
990 }
991 ultimately AOT_have ‹(O!x & O!x & □∀F([F]x ≡ [F]x)) ∨ (A!x & A!x & □∀F(x[F] ≡ x[F]))›
992 using "oa-exist:3" "∨I"(1) "∨I"(2) "∨E"(3) "raa-cor:1" by blast
993 AOT_thus ‹x = x›
994 using "identity:1"[THEN "df-rules-formulas[4]"] "→E" by blast
995 }
996qed
997
998instance rel :: ("{AOT_κs,AOT_Term_id_2}") AOT_Term_id_2
999proof
1000 AOT_modally_strict {
1001 fix F :: "<'a> AOT_var"
1002 AOT_have 0: ‹[λx⇩1...x⇩n [F]x⇩1...x⇩n] = F›
1003 by (simp add: "lambda-predicates:3"[axiom_inst])
1004 AOT_have ‹[λx⇩1...x⇩n [F]x⇩1...x⇩n]↓›
1005 by "cqt:2[lambda]"
1006 AOT_hence ‹[λx⇩1...x⇩n [F]x⇩1...x⇩n] = [λx⇩1...x⇩n [F]x⇩1...x⇩n]›
1007 using "lambda-predicates:1"[axiom_inst] "→E" by blast
1008 AOT_show ‹F = F› using "rule=E" 0 by force
1009 }
1010qed
1011
1012instance 𝗈 :: AOT_Term_id_2
1013proof
1014 AOT_modally_strict {
1015 fix p
1016 AOT_have 0: ‹[λ p] = p›
1017 by (simp add: "lambda-predicates:3[zero]"[axiom_inst])
1018 AOT_have ‹[λ p]↓›
1019 by (rule "cqt:2[lambda0]"[axiom_inst])
1020 AOT_hence ‹[λ p] = [λ p]›
1021 using "lambda-predicates:1[zero]"[axiom_inst] "→E" by blast
1022 AOT_show ‹p = p› using "rule=E" 0 by force
1023 }
1024qed
1025
1026instance prod :: (AOT_Term_id_2, AOT_Term_id_2) AOT_Term_id_2
1027proof
1028 AOT_modally_strict {
1029 fix α :: ‹('a×'b) AOT_var›
1030 AOT_show ‹α = α›
1031 proof (induct)
1032 AOT_show ‹τ = τ› if ‹τ↓› for τ :: ‹'a×'b›
1033 using that
1034 proof (induct τ)
1035 fix τ⇩1 :: 'a and τ⇩2 :: 'b
1036 AOT_assume ‹«(τ⇩1,τ⇩2)»↓›
1037 AOT_hence ‹τ⇩1↓› and ‹τ⇩2↓› using "≡⇩d⇩fE" "&E" tuple_denotes by blast+
1038 AOT_hence ‹τ⇩1 = τ⇩1› and ‹τ⇩2 = τ⇩2› using "id-eq:1"[unvarify α] by blast+
1039 AOT_thus ‹«(τ⇩1, τ⇩2)» = «(τ⇩1, τ⇩2)»› by (metis "≡⇩d⇩fI" "&I" tuple_identity_1)
1040 qed
1041 qed
1042 }
1043qed
1044
1045AOT_register_type_constraints
1046 Term: ‹_::AOT_Term_id_2› ‹_::AOT_Term_id_2›
1047AOT_register_type_constraints
1048 Individual: ‹κ› ‹_::{AOT_κs, AOT_Term_id_2}›
1049AOT_register_type_constraints
1050 Relation: ‹<_::{AOT_κs, AOT_Term_id_2}>›
1051
1052
1053AOT_theorem "id-eq:2": ‹α = β → β = α›
1054
1064
1065proof (rule "→I")
1066 AOT_assume ‹α = β›
1067 moreover AOT_have ‹β = β› using calculation "rule=E"[of _ "λ τ . «τ = β»" "AOT_term_of_var α" "AOT_term_of_var β"] by blast
1068 moreover AOT_have ‹α = α → α = α› using "if-p-then-p" by blast
1069 ultimately AOT_show ‹β = α›
1070 using "→E" "→I" "rule=E"[of _ "λ τ . «(τ = τ) → (τ = α)»" "AOT_term_of_var α" "AOT_term_of_var β"] by blast
1071qed
1072
1073AOT_theorem "id-eq:3": ‹α = β & β = γ → α = γ›
1074 using "rule=E" "→I" "&E" by blast
1075
1076AOT_theorem "id-eq:4": ‹α = β ≡ ∀γ (α = γ ≡ β = γ)›
1077proof (rule "≡I"; rule "→I")
1078 AOT_assume 0: ‹α = β›
1079 AOT_hence 1: ‹β = α› using "id-eq:2" "→E" by blast
1080 AOT_show ‹∀γ (α = γ ≡ β = γ)›
1081 by (rule GEN) (metis "≡I" "→I" 0 "1" "rule=E")
1082next
1083 AOT_assume ‹∀γ (α = γ ≡ β = γ)›
1084 AOT_hence ‹α = α ≡ β = α› using "∀E"(2) by blast
1085 AOT_hence ‹α = α → β = α› using "≡E"(1) "→I" by blast
1086 AOT_hence ‹β = α› using "id-eq:1" "→E" by blast
1087 AOT_thus ‹α = β› using "id-eq:2" "→E" by blast
1088qed
1089
1090AOT_theorem "rule=I:1": assumes ‹τ↓› shows ‹τ = τ›
1091proof -
1092 AOT_have ‹∀α (α = α)›
1093 by (rule GEN) (metis "id-eq:1")
1094 AOT_thus ‹τ = τ› using assms "∀E" by blast
1095qed
1096
1097AOT_theorem "rule=I:2[const_var]": "α = α"
1098 using "id-eq:1".
1099
1100AOT_theorem "rule=I:2[lambda]": assumes ‹INSTANCE_OF_CQT_2(φ)› shows "[λν⇩1...ν⇩n φ{ν⇩1...ν⇩n}] = [λν⇩1...ν⇩n φ{ν⇩1...ν⇩n}]"
1101proof -
1102 AOT_have ‹∀α (α = α)›
1103 by (rule GEN) (metis "id-eq:1")
1104 moreover AOT_have ‹[λν⇩1...ν⇩n φ{ν⇩1...ν⇩n}]↓› using assms by (rule "cqt:2[lambda]"[axiom_inst])
1105 ultimately AOT_show ‹[λν⇩1...ν⇩n φ{ν⇩1...ν⇩n}] = [λν⇩1...ν⇩n φ{ν⇩1...ν⇩n}]› using assms "∀E" by blast
1106qed
1107
1108lemmas "=I" = "rule=I:1" "rule=I:2[const_var]" "rule=I:2[lambda]"
1109
1110AOT_theorem "rule-id-df:1":
1111 assumes ‹τ{α⇩1...α⇩n} =⇩d⇩f σ{α⇩1...α⇩n}› and ‹σ{τ⇩1...τ⇩n}↓›
1112 shows ‹τ{τ⇩1...τ⇩n} = σ{τ⇩1...τ⇩n}›
1113proof -
1114 AOT_have ‹σ{τ⇩1...τ⇩n}↓ → τ{τ⇩1...τ⇩n} = σ{τ⇩1...τ⇩n}›
1115 using "df-rules-terms[3]" assms(1) "&E" by blast
1116 AOT_thus ‹τ{τ⇩1...τ⇩n} = σ{τ⇩1...τ⇩n}›
1117 using assms(2) "→E" by blast
1118qed
1119
1120AOT_theorem "rule-id-df:1[zero]":
1121 assumes ‹τ =⇩d⇩f σ› and ‹σ↓›
1122 shows ‹τ = σ›
1123proof -
1124 AOT_have ‹σ↓ → τ = σ›
1125 using "df-rules-terms[4]" assms(1) "&E" by blast
1126 AOT_thus ‹τ = σ›
1127 using assms(2) "→E" by blast
1128qed
1129
1130AOT_theorem "rule-id-df:2:a":
1131 assumes ‹τ{α⇩1...α⇩n} =⇩d⇩f σ{α⇩1...α⇩n}› and ‹σ{τ⇩1...τ⇩n}↓› and ‹φ{τ{τ⇩1...τ⇩n}}›
1132 shows ‹φ{σ{τ⇩1...τ⇩n}}›
1133proof -
1134 AOT_have ‹τ{τ⇩1...τ⇩n} = σ{τ⇩1...τ⇩n}› using "rule-id-df:1" assms(1,2) by blast
1135 AOT_thus ‹φ{σ{τ⇩1...τ⇩n}}› using assms(3) "rule=E" by blast
1136qed
1137
1138
1139AOT_theorem "rule-id-df:2:a[2]":
1140 assumes ‹τ{«(α⇩1,α⇩2)»} =⇩d⇩f σ{«(α⇩1,α⇩2)»}› and ‹σ{«(τ⇩1,τ⇩2)»}↓› and ‹φ{τ{«(τ⇩1,τ⇩2)»}}›
1141 shows ‹φ{σ{«(τ⇩1,τ⇩2)»}}›
1142proof -
1143 AOT_have ‹τ{«(τ⇩1,τ⇩2)»} = σ{«(τ⇩1,τ⇩2)»}›
1144 proof -
1145 AOT_have ‹σ{«(τ⇩1,τ⇩2)»}↓ → τ{«(τ⇩1,τ⇩2)»} = σ{«(τ⇩1,τ⇩2)»}›
1146 using assms by (simp add: AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def)
1147 AOT_thus ‹τ{«(τ⇩1,τ⇩2)»} = σ{«(τ⇩1,τ⇩2)»}›
1148 using assms(2) "→E" by blast
1149 qed
1150 AOT_thus ‹φ{σ{«(τ⇩1,τ⇩2)»}}› using assms(3) "rule=E" by blast
1151qed
1152
1153AOT_theorem "rule-id-df:2:a[zero]":
1154 assumes ‹τ =⇩d⇩f σ› and ‹σ↓› and ‹φ{τ}›
1155 shows ‹φ{σ}›
1156proof -
1157 AOT_have ‹τ = σ› using "rule-id-df:1[zero]" assms(1,2) by blast
1158 AOT_thus ‹φ{σ}› using assms(3) "rule=E" by blast
1159qed
1160
1161lemmas "=⇩d⇩fE" = "rule-id-df:2:a" "rule-id-df:2:a[zero]"
1162
1163AOT_theorem "rule-id-df:2:b":
1164 assumes ‹τ{α⇩1...α⇩n} =⇩d⇩f σ{α⇩1...α⇩n}› and ‹σ{τ⇩1...τ⇩n}↓› and ‹φ{σ{τ⇩1...τ⇩n}}›
1165 shows ‹φ{τ{τ⇩1...τ⇩n}}›
1166proof -
1167 AOT_have ‹τ{τ⇩1...τ⇩n} = σ{τ⇩1...τ⇩n}› using "rule-id-df:1" assms(1,2) by blast
1168 AOT_hence ‹σ{τ⇩1...τ⇩n} = τ{τ⇩1...τ⇩n}›
1169 using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1170 AOT_thus ‹φ{τ{τ⇩1...τ⇩n}}› using assms(3) "rule=E" by blast
1171qed
1172
1173
1174AOT_theorem "rule-id-df:2:b[2]":
1175 assumes ‹τ{«(α⇩1,α⇩2)»} =⇩d⇩f σ{«(α⇩1,α⇩2)»}› and ‹σ{«(τ⇩1,τ⇩2)»}↓› and ‹φ{σ{«(τ⇩1,τ⇩2)»}}›
1176 shows ‹φ{τ{«(τ⇩1,τ⇩2)»}}›
1177proof -
1178 AOT_have ‹τ{«(τ⇩1,τ⇩2)»} = σ{«(τ⇩1,τ⇩2)»}›
1179 proof -
1180 AOT_have ‹σ{«(τ⇩1,τ⇩2)»}↓ → τ{«(τ⇩1,τ⇩2)»} = σ{«(τ⇩1,τ⇩2)»}›
1181 using assms by (simp add: AOT_sem_conj AOT_sem_imp AOT_sem_eq AOT_sem_not AOT_sem_denotes AOT_model_id_def)
1182 AOT_thus ‹τ{«(τ⇩1,τ⇩2)»} = σ{«(τ⇩1,τ⇩2)»}›
1183 using assms(2) "→E" by blast
1184 qed
1185 AOT_hence ‹σ{«(τ⇩1,τ⇩2)»} = τ{«(τ⇩1,τ⇩2)»}›
1186 using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1187 AOT_thus ‹φ{τ{«(τ⇩1,τ⇩2)»}}› using assms(3) "rule=E" by blast
1188qed
1189
1190AOT_theorem "rule-id-df:2:b[zero]":
1191 assumes ‹τ =⇩d⇩f σ› and ‹σ↓› and ‹φ{σ}›
1192 shows ‹φ{τ}›
1193proof -
1194 AOT_have ‹τ = σ› using "rule-id-df:1[zero]" assms(1,2) by blast
1195 AOT_hence ‹σ = τ›
1196 using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
1197 AOT_thus ‹φ{τ}› using assms(3) "rule=E" by blast
1198qed
1199
1200lemmas "=⇩d⇩fI" = "rule-id-df:2:b" "rule-id-df:2:b[zero]"
1201
1202AOT_theorem "free-thms:1": ‹τ↓ ≡ ∃β (β = τ)›
1203 by (metis "∃E" "rule=I:1" "t=t-proper:2" "→I" "∃I"(1) "≡I" "→E")
1204
1205AOT_theorem "free-thms:2": ‹∀α φ{α} → (∃β (β = τ) → φ{τ})›
1206 by (metis "∃E" "rule=E" "cqt:2[const_var]"[axiom_inst] "→I" "∀E"(1))
1207
1208AOT_theorem "free-thms:3[const_var]": ‹∃β (β = α)›
1209 by (meson "∃I"(2) "id-eq:1")
1210
1211AOT_theorem "free-thms:3[lambda]": assumes ‹INSTANCE_OF_CQT_2(φ)› shows ‹∃β (β = [λν⇩1...ν⇩n φ{ν⇩1...ν⇩n}])›
1212 by (meson "=I"(3) assms "cqt:2[lambda]"[axiom_inst] "existential:1")
1213
1214AOT_theorem "free-thms:4[rel]": ‹([Π]κ⇩1...κ⇩n ∨ κ⇩1...κ⇩n[Π]) → ∃β (β = Π)›
1215 by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1216
1217
1219AOT_theorem "free-thms:4[vars]": ‹([Π]κ⇩1...κ⇩n ∨ κ⇩1...κ⇩n[Π]) → ∃β⇩1...∃β⇩n (β⇩1...β⇩n = κ⇩1...κ⇩n)›
1220 by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1221
1222AOT_theorem "free-thms:4[1,rel]": ‹([Π]κ ∨ κ[Π]) → ∃β (β = Π)›
1223 by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1224AOT_theorem "free-thms:4[1,1]": ‹([Π]κ ∨ κ[Π]) → ∃β (β = κ)›
1225 by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst] "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
1226
1227AOT_theorem "free-thms:4[2,rel]": ‹([Π]κ⇩1κ⇩2 ∨ κ⇩1κ⇩2[Π]) → ∃β (β = Π)›
1228 by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1229AOT_theorem "free-thms:4[2,1]": ‹([Π]κ⇩1κ⇩2 ∨ κ⇩1κ⇩2[Π]) → ∃β (β = κ⇩1)›
1230 by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1231AOT_theorem "free-thms:4[2,2]": ‹([Π]κ⇩1κ⇩2 ∨ κ⇩1κ⇩2[Π]) → ∃β (β = κ⇩2)›
1232 by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[2]"[axiom_inst] "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
1233AOT_theorem "free-thms:4[3,rel]": ‹([Π]κ⇩1κ⇩2κ⇩3 ∨ κ⇩1κ⇩2κ⇩3[Π]) → ∃β (β = Π)›
1234 by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1235AOT_theorem "free-thms:4[3,1]": ‹([Π]κ⇩1κ⇩2κ⇩3 ∨ κ⇩1κ⇩2κ⇩3[Π]) → ∃β (β = κ⇩1)›
1236 by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1237AOT_theorem "free-thms:4[3,2]": ‹([Π]κ⇩1κ⇩2κ⇩3 ∨ κ⇩1κ⇩2κ⇩3[Π]) → ∃β (β = κ⇩2)›
1238 by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1239AOT_theorem "free-thms:4[3,3]": ‹([Π]κ⇩1κ⇩2κ⇩3 ∨ κ⇩1κ⇩2κ⇩3[Π]) → ∃β (β = κ⇩3)›
1240 by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[3]"[axiom_inst] "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
1241AOT_theorem "free-thms:4[4,rel]": ‹([Π]κ⇩1κ⇩2κ⇩3κ⇩4 ∨ κ⇩1κ⇩2κ⇩3κ⇩4[Π]) → ∃β (β = Π)›
1242 by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1243AOT_theorem "free-thms:4[4,1]": ‹([Π]κ⇩1κ⇩2κ⇩3κ⇩4 ∨ κ⇩1κ⇩2κ⇩3κ⇩4[Π]) → ∃β (β = κ⇩1)›
1244 by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1245AOT_theorem "free-thms:4[4,2]": ‹([Π]κ⇩1κ⇩2κ⇩3κ⇩4 ∨ κ⇩1κ⇩2κ⇩3κ⇩4[Π]) → ∃β (β = κ⇩2)›
1246 by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1247AOT_theorem "free-thms:4[4,3]": ‹([Π]κ⇩1κ⇩2κ⇩3κ⇩4 ∨ κ⇩1κ⇩2κ⇩3κ⇩4[Π]) → ∃β (β = κ⇩3)›
1248 by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1249AOT_theorem "free-thms:4[4,4]": ‹([Π]κ⇩1κ⇩2κ⇩3κ⇩4 ∨ κ⇩1κ⇩2κ⇩3κ⇩4[Π]) → ∃β (β = κ⇩4)›
1250 by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[4]"[axiom_inst] "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
1251
1252AOT_theorem "ex:1:a": ‹∀α α↓›
1253 by (rule GEN) (fact "cqt:2[const_var]"[axiom_inst])
1254AOT_theorem "ex:1:b": ‹∀α∃β(β = α)›
1255 by (rule GEN) (fact "free-thms:3[const_var]")
1256
1257AOT_theorem "ex:2:a": ‹□α↓›
1258 by (rule RN) (fact "cqt:2[const_var]"[axiom_inst])
1259AOT_theorem "ex:2:b": ‹□∃β(β = α)›
1260 by (rule RN) (fact "free-thms:3[const_var]")
1261
1262AOT_theorem "ex:3:a": ‹□∀α α↓›
1263 by (rule RN) (fact "ex:1:a")
1264AOT_theorem "ex:3:b": ‹□∀α∃β(β = α)›
1265 by (rule RN) (fact "ex:1:b")
1266
1267AOT_theorem "ex:4:a": ‹∀α □α↓›
1268 by (rule GEN; rule RN) (fact "cqt:2[const_var]"[axiom_inst])
1269AOT_theorem "ex:4:b": ‹∀α□∃β(β = α)›
1270 by (rule GEN; rule RN) (fact "free-thms:3[const_var]")
1271
1272AOT_theorem "ex:5:a": ‹□∀α □α↓›
1273 by (rule RN) (simp add: "ex:4:a")
1274AOT_theorem "ex:5:b": ‹□∀α□∃β(β = α)›
1275 by (rule RN) (simp add: "ex:4:b")
1276
1277AOT_theorem "all-self=:1": ‹□∀α(α = α)›
1278 by (rule RN; rule GEN) (fact "id-eq:1")
1279AOT_theorem "all-self=:2": ‹∀α□(α = α)›
1280 by (rule GEN; rule RN) (fact "id-eq:1")
1281
1282AOT_theorem "id-nec:1": ‹α = β → □(α = β)›
1283proof(rule "→I")
1284 AOT_assume ‹α = β›
1285 moreover AOT_have ‹□(α = α)›
1286 by (rule RN) (fact "id-eq:1")
1287 ultimately AOT_show ‹□(α = β)› using "rule=E" by fast
1288qed
1289
1290AOT_theorem "id-nec:2": ‹τ = σ → □(τ = σ)›
1291proof(rule "→I")
1292 AOT_assume asm: ‹τ = σ›
1293 moreover AOT_have ‹τ↓›
1294 using calculation "t=t-proper:1" "→E" by blast
1295 moreover AOT_have ‹□(τ = τ)›
1296 using calculation "all-self=:2" "∀E"(1) by blast
1297 ultimately AOT_show ‹□(τ = σ)› using "rule=E" by fast
1298qed
1299
1300AOT_theorem "term-out:1": ‹φ{α} ≡ ∃β (β = α & φ{β})›
1301proof (rule "≡I"; rule "→I")
1302 AOT_assume asm: ‹φ{α}›
1303 AOT_show ‹∃β (β = α & φ{β})›
1304 by (rule "∃I"(2)[where β=α]; rule "&I")
1305 (auto simp: "id-eq:1" asm)
1306next
1307 AOT_assume 0: ‹∃β (β = α & φ{β})›
1308
1310 AOT_obtain β where ‹β = α & φ{β}› using "instantiation"[rotated, OF 0] by blast
1311 AOT_thus ‹φ{α}› using "&E" "rule=E" by blast
1312qed
1313
1314AOT_theorem "term-out:2": ‹τ↓ → (φ{τ} ≡ ∃α(α = τ & φ{α}))›
1315proof(rule "→I")
1316 AOT_assume ‹τ↓›
1317 moreover AOT_have ‹∀α (φ{α} ≡ ∃β (β = α & φ{β}))›
1318 by (rule GEN) (fact "term-out:1")
1319 ultimately AOT_show ‹φ{τ} ≡ ∃α(α = τ & φ{α})›
1320 using "∀E" by blast
1321qed
1322
1323
1324AOT_theorem "term-out:3": ‹(φ{α} & ∀β(φ{β} → β = α)) ≡ ∀β(φ{β} ≡ β = α)›
1325 apply (rule "≡I"; rule "→I")
1326 apply (frule "&E"(1)) apply (drule "&E"(2))
1327 apply (rule GEN; rule "≡I"; rule "→I")
1328 using "rule-ui:2[const_var]" "vdash-properties:5" apply blast
1329 apply (meson "rule=E" "id-eq:1")
1330 apply (rule "&I")
1331 using "id-eq:1" "≡E"(2) "rule-ui:3" apply blast
1332 apply (rule GEN; rule "→I")
1333 using "≡E"(1) "rule-ui:2[const_var]" by blast
1334
1335AOT_theorem "term-out:4": ‹(φ{β} & ∀α(φ{α} → α = β)) ≡ ∀α(φ{α} ≡ α = β)›
1336 using "term-out:3" .
1337
1338
1339AOT_define AOT_exists_unique :: ‹α ⇒ φ ⇒ φ›
1340 "uniqueness:1": ‹«AOT_exists_unique φ» ≡⇩d⇩f ∃α (φ{α} & ∀β (φ{β} → β = α))›
1341syntax "_AOT_exists_unique" :: ‹α ⇒ φ ⇒ φ› ("∃!_ _" [1,40])
1342AOT_syntax_print_translations
1343 "_AOT_exists_unique τ φ" <= "CONST AOT_exists_unique (_abs τ φ)"
1344syntax
1345 "_AOT_exists_unique_ellipse" :: ‹id_position ⇒ id_position ⇒ φ ⇒ φ› (‹∃!_...∃!_ _› [1,40])
1346parse_ast_translation‹[(\<^syntax_const>‹_AOT_exists_unique_ellipse›, fn ctx => fn [a,b,c] =>
1347 Ast.mk_appl (Ast.Constant "AOT_exists_unique") [parseEllipseList "_AOT_vars" ctx [a,b],c]),
1348(\<^syntax_const>‹_AOT_exists_unique›,AOT_restricted_binder \<^const_name>‹AOT_exists_unique› \<^const_syntax>‹AOT_conj›)]›
1349print_translation‹AOT_syntax_print_translations
1350 [AOT_preserve_binder_abs_tr' \<^const_syntax>‹AOT_exists_unique› \<^syntax_const>‹_AOT_exists_unique› (\<^syntax_const>‹_AOT_exists_unique_ellipse›, true) \<^const_name>‹AOT_conj›,
1351 AOT_binder_trans @{theory} @{binding "AOT_exists_unique_binder"} \<^syntax_const>‹_AOT_exists_unique›]
1352›
1353
1354
1355context AOT_meta_syntax
1356begin
1357notation AOT_exists_unique (binder "❙∃❙!" 20)
1358end
1359context AOT_no_meta_syntax
1360begin
1361no_notation AOT_exists_unique (binder "❙∃❙!" 20)
1362end
1363
1364AOT_theorem "uniqueness:2": ‹∃!α φ{α} ≡ ∃α∀β(φ{β} ≡ β = α)›
1365proof(rule "≡I"; rule "→I")
1366 AOT_assume ‹∃!α φ{α}›
1367 AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
1368 using "uniqueness:1" "≡⇩d⇩fE" by blast
1369 then AOT_obtain α where ‹φ{α} & ∀β (φ{β} → β = α)› using "instantiation"[rotated] by blast
1370 AOT_hence ‹∀β(φ{β} ≡ β = α)› using "term-out:3" "≡E" by blast
1371 AOT_thus ‹∃α∀β(φ{β} ≡ β = α)›
1372 using "∃I" by fast
1373next
1374 AOT_assume ‹∃α∀β(φ{β} ≡ β = α)›
1375 then AOT_obtain α where ‹∀β (φ{β} ≡ β = α)› using "instantiation"[rotated] by blast
1376 AOT_hence ‹φ{α} & ∀β (φ{β} → β = α)› using "term-out:3" "≡E" by blast
1377 AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
1378 using "∃I" by fast
1379 AOT_thus ‹∃!α φ{α}›
1380 using "uniqueness:1" "≡⇩d⇩fI" by blast
1381qed
1382
1383AOT_theorem "uni-most": ‹∃!α φ{α} → ∀β∀γ((φ{β} & φ{γ}) → β = γ)›
1384proof(rule "→I"; rule GEN; rule GEN; rule "→I")
1385 fix β γ
1386 AOT_assume ‹∃!α φ{α}›
1387 AOT_hence ‹∃α∀β(φ{β} ≡ β = α)›
1388 using "uniqueness:2" "≡E" by blast
1389 then AOT_obtain α where ‹∀β(φ{β} ≡ β = α)›
1390 using "instantiation"[rotated] by blast
1391 moreover AOT_assume ‹φ{β} & φ{γ}›
1392 ultimately AOT_have ‹β = α› and ‹γ = α›
1393 using "∀E"(2) "&E" "≡E"(1,2) by blast+
1394 AOT_thus ‹β = γ›
1395 by (metis "rule=E" "id-eq:2" "→E")
1396qed
1397
1398AOT_theorem "nec-exist-!": ‹∀α(φ{α} → □φ{α}) → (∃!α φ{α} → ∃!α □φ{α})›
1399proof (rule "→I"; rule "→I")
1400 AOT_assume a: ‹∀α(φ{α} → □φ{α})›
1401 AOT_assume ‹∃!α φ{α}›
1402 AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))› using "uniqueness:1" "≡⇩d⇩fE" by blast
1403 then AOT_obtain α where ξ: ‹φ{α} & ∀β (φ{β} → β = α)› using "instantiation"[rotated] by blast
1404 AOT_have ‹□φ{α}›
1405 using ξ a "&E" "∀E" "→E" by fast
1406 moreover AOT_have ‹∀β (□φ{β} → β = α)›
1407 apply (rule GEN; rule "→I")
1408 using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
1409 ultimately AOT_have ‹(□φ{α} & ∀β (□φ{β} → β = α))›
1410 using "&I" by blast
1411 AOT_thus ‹∃!α □φ{α}›
1412 using "uniqueness:1" "≡⇩d⇩fI" "∃I" by fast
1413qed
1414
1415AOT_theorem "act-cond": ‹❙𝒜(φ → ψ) → (❙𝒜φ → ❙𝒜ψ)›
1416 using "→I" "≡E"(1) "logic-actual-nec:2"[axiom_inst] by blast
1417
1418AOT_theorem "nec-imp-act": ‹□φ → ❙𝒜φ›
1419 by (metis "act-cond" "contraposition:1[2]" "≡E"(4) "qml:2"[THEN act_closure, axiom_inst] "qml-act:2"[axiom_inst] RAA(1) "→E" "→I")
1420
1421AOT_theorem "act-conj-act:1": ‹❙𝒜(❙𝒜φ → φ)›
1422 using "→I" "≡E"(2) "logic-actual-nec:2"[axiom_inst] "logic-actual-nec:4"[axiom_inst] by blast
1423
1424AOT_theorem "act-conj-act:2": ‹❙𝒜(φ → ❙𝒜φ)›
1425 by (metis "→I" "≡E"(2, 4) "logic-actual-nec:2"[axiom_inst] "logic-actual-nec:4"[axiom_inst] RAA(1))
1426
1427AOT_theorem "act-conj-act:3": ‹(❙𝒜φ & ❙𝒜ψ) → ❙𝒜(φ & ψ)›
1428proof -
1429 AOT_have ‹□(φ → (ψ → (φ & ψ)))›
1430 by (rule RN) (fact Adjunction)
1431 AOT_hence ‹❙𝒜(φ → (ψ → (φ & ψ)))›
1432 using "nec-imp-act" "→E" by blast
1433 AOT_hence ‹❙𝒜φ → ❙𝒜(ψ → (φ & ψ))›
1434 using "act-cond" "→E" by blast
1435 moreover AOT_have ‹❙𝒜(ψ → (φ & ψ)) → (❙𝒜ψ → ❙𝒜(φ & ψ))›
1436 by (fact "act-cond")
1437 ultimately AOT_have ‹❙𝒜φ → (❙𝒜ψ → ❙𝒜(φ & ψ))›
1438 using "→I" "→E" by metis
1439 AOT_thus ‹(❙𝒜φ & ❙𝒜ψ) → ❙𝒜(φ & ψ)›
1440 by (metis Importation "→E")
1441qed
1442
1443AOT_theorem "act-conj-act:4": ‹❙𝒜(❙𝒜φ ≡ φ)›
1444proof -
1445 AOT_have ‹(❙𝒜(❙𝒜φ → φ) & ❙𝒜(φ → ❙𝒜φ)) → ❙𝒜((❙𝒜φ → φ) & (φ → ❙𝒜φ))›
1446 by (fact "act-conj-act:3")
1447 moreover AOT_have ‹❙𝒜(❙𝒜φ → φ) & ❙𝒜(φ → ❙𝒜φ)›
1448 using "&I" "act-conj-act:1" "act-conj-act:2" by simp
1449 ultimately AOT_have ζ: ‹❙𝒜((❙𝒜φ → φ) & (φ → ❙𝒜φ))›
1450 using "→E" by blast
1451 AOT_have ‹❙𝒜(((❙𝒜φ → φ) & (φ → ❙𝒜φ)) → (❙𝒜φ ≡ φ))›
1452 using "conventions:3"[THEN "df-rules-formulas[2]", THEN act_closure, axiom_inst] by blast
1453 AOT_hence ‹❙𝒜((❙𝒜φ → φ) & (φ → ❙𝒜φ)) → ❙𝒜(❙𝒜φ ≡ φ)›
1454 using "act-cond" "→E" by blast
1455 AOT_thus ‹❙𝒜(❙𝒜φ ≡ φ)› using ζ "→E" by blast
1456qed
1457
1458
1459inductive arbitrary_actualization for φ where
1460 ‹arbitrary_actualization φ «❙𝒜φ»›
1461| ‹arbitrary_actualization φ «❙𝒜ψ»› if ‹arbitrary_actualization φ ψ›
1462declare arbitrary_actualization.cases[AOT] arbitrary_actualization.induct[AOT]
1463 arbitrary_actualization.simps[AOT] arbitrary_actualization.intros[AOT]
1464syntax arbitrary_actualization :: ‹φ' ⇒ φ' ⇒ AOT_prop› ("ARBITRARY'_ACTUALIZATION'(_,_')")
1465
1466notepad
1467begin
1468 AOT_modally_strict {
1469 fix φ
1470 AOT_have ‹ARBITRARY_ACTUALIZATION(❙𝒜φ ≡ φ, ❙𝒜(❙𝒜φ ≡ φ))›
1471 using AOT_PLM.arbitrary_actualization.intros by metis
1472 AOT_have ‹ARBITRARY_ACTUALIZATION(❙𝒜φ ≡ φ, ❙𝒜❙𝒜(❙𝒜φ ≡ φ))›
1473 using AOT_PLM.arbitrary_actualization.intros by metis
1474 AOT_have ‹ARBITRARY_ACTUALIZATION(❙𝒜φ ≡ φ, ❙𝒜❙𝒜❙𝒜(❙𝒜φ ≡ φ))›
1475 using AOT_PLM.arbitrary_actualization.intros by metis
1476 }
1477end
1478
1479
1480AOT_theorem "closure-act:1": assumes ‹ARBITRARY_ACTUALIZATION(❙𝒜φ ≡ φ, ψ)› shows ‹ψ›
1481using assms proof(induct)
1482 case 1
1483 AOT_show ‹❙𝒜(❙𝒜φ ≡ φ)›
1484 by (simp add: "act-conj-act:4")
1485next
1486 case (2 ψ)
1487 AOT_thus ‹❙𝒜ψ›
1488 by (metis arbitrary_actualization.simps "≡E"(1) "logic-actual-nec:4"[axiom_inst])
1489qed
1490
1491AOT_theorem "closure-act:2": ‹∀α ❙𝒜(❙𝒜φ{α} ≡ φ{α})›
1492 by (simp add: "act-conj-act:4" "∀I")
1493
1494AOT_theorem "closure-act:3": ‹❙𝒜∀α ❙𝒜(❙𝒜φ{α} ≡ φ{α})›
1495 by (metis (no_types, lifting) "act-conj-act:4" "≡E"(1,2) "logic-actual-nec:3"[axiom_inst] "logic-actual-nec:4"[axiom_inst] "∀I")
1496
1497AOT_theorem "closure-act:4": ‹❙𝒜∀α⇩1...∀α⇩n ❙𝒜(❙𝒜φ{α⇩1...α⇩n} ≡ φ{α⇩1...α⇩n})›
1498 using "closure-act:3" .
1499
1500
1501AOT_theorem "RA[1]": assumes ‹❙⊢ φ› shows ‹❙⊢ ❙𝒜φ›
1502
1503 using "¬¬E" assms "≡E"(3) "logic-actual"[act_axiom_inst] "logic-actual-nec:1"[axiom_inst] "modus-tollens:2" by blast
1504AOT_theorem "RA[2]": assumes ‹❙⊢⇩□ φ› shows ‹❙𝒜φ›
1505
1506 using RN assms "nec-imp-act" "vdash-properties:5" by blast
1507AOT_theorem "RA[3]": assumes ‹Γ ❙⊢⇩□ φ› shows ‹❙𝒜Γ ❙⊢⇩□ ❙𝒜φ›
1508 using assms by (meson AOT_sem_act imageI)
1509
1510
1511AOT_act_theorem "ANeg:1": ‹¬❙𝒜φ ≡ ¬φ›
1512 by (simp add: "RA[1]" "contraposition:1[1]" "deduction-theorem" "≡I" "logic-actual"[act_axiom_inst])
1513
1514AOT_act_theorem "ANeg:2": ‹¬❙𝒜¬φ ≡ φ›
1515 using "ANeg:1" "≡I" "≡E"(5) "useful-tautologies:1" "useful-tautologies:2" by blast
1516
1517AOT_theorem "Act-Basic:1": ‹❙𝒜φ ∨ ❙𝒜¬φ›
1518 by (meson "∨I"(1,2) "≡E"(2) "logic-actual-nec:1"[axiom_inst] "raa-cor:1")
1519
1520AOT_theorem "Act-Basic:2": ‹❙𝒜(φ & ψ) ≡ (❙𝒜φ & ❙𝒜ψ)›
1521proof (rule "≡I"; rule "→I")
1522 AOT_assume ‹❙𝒜(φ & ψ)›
1523 moreover AOT_have ‹❙𝒜((φ & ψ) → φ)›
1524 by (simp add: "RA[2]" "Conjunction Simplification"(1))
1525 moreover AOT_have ‹❙𝒜((φ & ψ) → ψ)›
1526 by (simp add: "RA[2]" "Conjunction Simplification"(2))
1527 ultimately AOT_show ‹❙𝒜φ & ❙𝒜ψ›
1528 using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
1529next
1530 AOT_assume ‹❙𝒜φ & ❙𝒜ψ›
1531 AOT_thus ‹❙𝒜(φ & ψ)›
1532 using "act-conj-act:3" "vdash-properties:6" by blast
1533qed
1534
1535AOT_theorem "Act-Basic:3": ‹❙𝒜(φ ≡ ψ) ≡ (❙𝒜(φ → ψ) & ❙𝒜(ψ → φ))›
1536proof (rule "≡I"; rule "→I")
1537 AOT_assume ‹❙𝒜(φ ≡ ψ)›
1538 moreover AOT_have ‹❙𝒜((φ ≡ ψ) → (φ → ψ))›
1539 by (simp add: "RA[2]" "deduction-theorem" "≡E"(1))
1540 moreover AOT_have ‹❙𝒜((φ ≡ ψ) → (ψ → φ))›
1541 by (simp add: "RA[2]" "deduction-theorem" "≡E"(2))
1542 ultimately AOT_show ‹❙𝒜(φ → ψ) & ❙𝒜(ψ → φ)›
1543 using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
1544next
1545 AOT_assume ‹❙𝒜(φ → ψ) & ❙𝒜(ψ → φ)›
1546 AOT_hence ‹❙𝒜((φ → ψ) & (ψ → φ))›
1547 by (metis "act-conj-act:3" "vdash-properties:10")
1548 moreover AOT_have ‹❙𝒜(((φ → ψ) & (ψ → φ)) → (φ ≡ ψ))›
1549 by (simp add: "conventions:3" "RA[2]" "df-rules-formulas[2]" "vdash-properties:1[2]")
1550 ultimately AOT_show ‹❙𝒜(φ ≡ ψ)›
1551 using "act-cond"[THEN "→E", THEN "→E"] by metis
1552qed
1553
1554AOT_theorem "Act-Basic:4": ‹(❙𝒜(φ → ψ) & ❙𝒜(ψ → φ)) ≡ (❙𝒜φ ≡ ❙𝒜ψ)›
1555proof (rule "≡I"; rule "→I")
1556 AOT_assume 0: ‹❙𝒜(φ → ψ) & ❙𝒜(ψ → φ)›
1557 AOT_show ‹❙𝒜φ ≡ ❙𝒜ψ›
1558 using 0 "&E" "act-cond"[THEN "→E", THEN "→E"] "≡I" "→I" by metis
1559next
1560 AOT_assume ‹❙𝒜φ ≡ ❙𝒜ψ›
1561 AOT_thus ‹❙𝒜(φ → ψ) & ❙𝒜(ψ → φ)›
1562 by (metis "→I" "logic-actual-nec:2"[axiom_inst] "≡E"(1,2) "&I")
1563qed
1564
1565AOT_theorem "Act-Basic:5": ‹❙𝒜(φ ≡ ψ) ≡ (❙𝒜φ ≡ ❙𝒜ψ)›
1566 using "Act-Basic:3" "Act-Basic:4" "≡E"(5) by blast
1567
1568AOT_theorem "Act-Basic:6": ‹❙𝒜φ ≡ □❙𝒜φ›
1569 by (simp add: "≡I" "qml:2"[axiom_inst] "qml-act:1"[axiom_inst])
1570
1571AOT_theorem "Act-Basic:7": ‹❙𝒜□φ → □❙𝒜φ›
1572 by (metis "Act-Basic:6" "→I" "→E" "≡E"(1,2) "nec-imp-act" "qml-act:2"[axiom_inst])
1573
1574AOT_theorem "Act-Basic:8": ‹□φ → □❙𝒜φ›
1575 using "Hypothetical Syllogism" "nec-imp-act" "qml-act:1"[axiom_inst] by blast
1576
1577AOT_theorem "Act-Basic:9": ‹❙𝒜(φ ∨ ψ) ≡ (❙𝒜φ ∨ ❙𝒜ψ)›
1578proof (rule "≡I"; rule "→I")
1579 AOT_assume ‹❙𝒜(φ ∨ ψ)›
1580 AOT_thus ‹❙𝒜φ ∨ ❙𝒜ψ›
1581 proof (rule "raa-cor:3")
1582 AOT_assume ‹¬(❙𝒜φ ∨ ❙𝒜ψ)›
1583 AOT_hence ‹¬❙𝒜φ & ¬❙𝒜ψ›
1584 by (metis "≡E"(1) "oth-class-taut:5:d")
1585 AOT_hence ‹❙𝒜¬φ & ❙𝒜¬ψ›
1586 using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] "&E" "&I" by metis
1587 AOT_hence ‹❙𝒜(¬φ & ¬ψ)›
1588 using "≡E" "Act-Basic:2" by metis
1589 moreover AOT_have ‹❙𝒜((¬φ & ¬ψ) ≡ ¬(φ ∨ ψ))›
1590 using "RA[2]" "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:d" by blast
1591 moreover AOT_have ‹❙𝒜(¬φ & ¬ψ) ≡ ❙𝒜(¬(φ ∨ ψ))›
1592 using calculation(2) by (metis "Act-Basic:5" "≡E"(1))
1593 ultimately AOT_have ‹❙𝒜(¬(φ ∨ ψ))› using "≡E" by blast
1594 AOT_thus ‹¬❙𝒜(φ ∨ ψ)›
1595 using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by auto
1596 qed
1597next
1598 AOT_assume ‹❙𝒜φ ∨ ❙𝒜ψ›
1599 AOT_thus ‹❙𝒜(φ ∨ ψ)›
1600 by (meson "RA[2]" "act-cond" "∨I"(1) "∨E"(1) "Disjunction Addition"(1) "Disjunction Addition"(2))
1601qed
1602
1603AOT_theorem "Act-Basic:10": ‹❙𝒜∃α φ{α} ≡ ∃α ❙𝒜φ{α}›
1604proof -
1605 AOT_have θ: ‹¬❙𝒜∀α ¬φ{α} ≡ ¬∀α ❙𝒜¬φ{α}›
1606 by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
1607 (metis "logic-actual-nec:3"[axiom_inst])
1608 AOT_have ξ: ‹¬∀α ❙𝒜¬φ{α} ≡ ¬∀α ¬❙𝒜φ{α}›
1609 by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
1610 (rule "logic-actual-nec:1"[THEN universal_closure, axiom_inst, THEN "cqt-basic:3"[THEN "→E"]])
1611 AOT_have ‹❙𝒜(∃α φ{α}) ≡ ❙𝒜(¬∀α ¬φ{α})›
1612 using "conventions:4"[THEN "df-rules-formulas[1]", THEN act_closure, axiom_inst]
1613 "conventions:4"[THEN "df-rules-formulas[2]", THEN act_closure, axiom_inst]
1614 "Act-Basic:4"[THEN "≡E"(1)] "&I" "Act-Basic:5"[THEN "≡E"(2)] by metis
1615 also AOT_have ‹… ≡ ¬❙𝒜∀α ¬φ{α}›
1616 by (simp add: "logic-actual-nec:1" "vdash-properties:1[2]")
1617 also AOT_have ‹… ≡ ¬∀α ❙𝒜 ¬φ{α}› using θ by blast
1618 also AOT_have ‹… ≡ ¬∀α ¬❙𝒜 φ{α}› using ξ by blast
1619 also AOT_have ‹… ≡ ∃α ❙𝒜 φ{α}›
1620 using "conventions:4"[THEN "≡Df"] by (metis "≡E"(6) "oth-class-taut:3:a")
1621 finally AOT_show ‹❙𝒜∃α φ{α} ≡ ∃α ❙𝒜φ{α}› .
1622qed
1623
1624
1625AOT_theorem "Act-Basic:11": ‹❙𝒜∀α(φ{α} ≡ ψ{α}) ≡ ∀α(❙𝒜φ{α} ≡ ❙𝒜ψ{α})›
1626proof(rule "≡I"; rule "→I")
1627 AOT_assume ‹❙𝒜∀α(φ{α} ≡ ψ{α})›
1628 AOT_hence ‹∀α❙𝒜(φ{α} ≡ ψ{α})›
1629 using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] by blast
1630 AOT_hence ‹❙𝒜(φ{α} ≡ ψ{α})› for α using "∀E" by blast
1631 AOT_hence ‹❙𝒜φ{α} ≡ ❙𝒜ψ{α}› for α by (metis "Act-Basic:5" "≡E"(1))
1632 AOT_thus ‹∀α(❙𝒜φ{α} ≡ ❙𝒜ψ{α})› by (rule "∀I")
1633next
1634 AOT_assume ‹∀α(❙𝒜φ{α} ≡ ❙𝒜ψ{α})›
1635 AOT_hence ‹❙𝒜φ{α} ≡ ❙𝒜ψ{α}› for α using "∀E" by blast
1636 AOT_hence ‹❙𝒜(φ{α} ≡ ψ{α})› for α by (metis "Act-Basic:5" "≡E"(2))
1637 AOT_hence ‹∀α ❙𝒜(φ{α} ≡ ψ{α})› by (rule "∀I")
1638 AOT_thus ‹❙𝒜∀α(φ{α} ≡ ψ{α})›
1639 using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
1640qed
1641
1642AOT_act_theorem "act-quant-uniq": ‹∀β(❙𝒜φ{β} ≡ β = α) ≡ ∀β(φ{β} ≡ β = α)›
1643proof(rule "≡I"; rule "→I")
1644 AOT_assume ‹∀β(❙𝒜φ{β} ≡ β = α)›
1645 AOT_hence ‹❙𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1646 AOT_hence ‹φ{β} ≡ β = α› for β
1647 using "≡I" "→I" "RA[1]" "≡E"(1) "≡E"(2) "logic-actual"[act_axiom_inst] "vdash-properties:6"
1648 by metis
1649 AOT_thus ‹∀β(φ{β} ≡ β = α)› by (rule "∀I")
1650next
1651 AOT_assume ‹∀β(φ{β} ≡ β = α)›
1652 AOT_hence ‹φ{β} ≡ β = α› for β using "∀E" by blast
1653 AOT_hence ‹❙𝒜φ{β} ≡ β = α› for β
1654 using "≡I" "→I" "RA[1]" "≡E"(1) "≡E"(2) "logic-actual"[act_axiom_inst] "vdash-properties:6"
1655 by metis
1656 AOT_thus ‹∀β(❙𝒜φ{β} ≡ β = α)› by (rule "∀I")
1657qed
1658
1659AOT_act_theorem "fund-cont-desc": ‹x = ❙ιx(φ{x}) ≡ ∀z(φ{z} ≡ z = x)›
1660 using descriptions[axiom_inst] "act-quant-uniq" "≡E"(5) by fast
1661
1662AOT_act_theorem hintikka: ‹x = ❙ιx(φ{x}) ≡ (φ{x} & ∀z (φ{z} → z = x))›
1663 using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" "fund-cont-desc" "≡E"(5) by blast
1664
1665
1666locale russel_axiom =
1667 fixes ψ
1668 assumes ψ_denotes_asm: "[v ⊨ ψ{κ}] ⟹ [v ⊨ κ↓]"
1669begin
1670AOT_act_theorem "russell-axiom": ‹ψ{❙ιx φ{x}} ≡ ∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})›
1671proof -
1672 AOT_have b: ‹∀x (x = ❙ιx φ{x} ≡ (φ{x} & ∀z(φ{z} → z = x)))›
1673 using hintikka "∀I" by fast
1674 show ?thesis
1675 proof(rule "≡I"; rule "→I")
1676 AOT_assume c: ‹ψ{❙ιx φ{x}}›
1677 AOT_hence d: ‹❙ιx φ{x}↓› using ψ_denotes_asm by blast
1678 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1679 then AOT_obtain a where a_def: ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1680 moreover AOT_have ‹a = ❙ιx φ{x} ≡ (φ{a} & ∀z(φ{z} → z = a))› using b "∀E" by blast
1681 ultimately AOT_have ‹φ{a} & ∀z(φ{z} → z = a)› using "≡E" by blast
1682 moreover AOT_have ‹ψ{a}›
1683 proof -
1684 AOT_have 1: ‹∀x∀y(x = y → y = x)›
1685 by (simp add: "id-eq:2" "universal-cor")
1686 AOT_have ‹a = ❙ιx φ{x} → ❙ιx φ{x} = a›
1687 by (rule "∀E"(1)[where τ="«❙ιx φ{x}»"]; rule "∀E"(2)[where β=a])
1688 (auto simp: 1 d "universal-cor")
1689 AOT_thus ‹ψ{a}›
1690 using a_def c "rule=E" "→E" by blast
1691 qed
1692 ultimately AOT_have ‹φ{a} & ∀z(φ{z} → z = a) & ψ{a}› by (rule "&I")
1693 AOT_thus ‹∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})› by (rule "∃I")
1694 next
1695 AOT_assume ‹∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})›
1696 then AOT_obtain b where g: ‹φ{b} & ∀z(φ{z} → z = b) & ψ{b}› using "instantiation"[rotated] by blast
1697 AOT_hence h: ‹b = ❙ιx φ{x} ≡ (φ{b} & ∀z(φ{z} → z = b))› using b "∀E" by blast
1698 AOT_have ‹φ{b} & ∀z(φ{z} → z = b)› and j: ‹ψ{b}› using g "&E" by blast+
1699 AOT_hence ‹b = ❙ιx φ{x}› using h "≡E" by blast
1700 AOT_thus ‹ψ{❙ιx φ{x}}› using j "rule=E" by blast
1701 qed
1702qed
1703end
1704
1705
1706
1708interpretation "russell-axiom[exe,1]": russel_axiom ‹λ κ . «[Π]κ»›
1709 by standard (metis "cqt:5:a[1]"[axiom_inst, THEN "→E"] "&E"(2))
1710interpretation "russell-axiom[exe,2,1,1]": russel_axiom ‹λ κ . «[Π]κκ'»›
1711 by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E")
1712interpretation "russell-axiom[exe,2,1,2]": russel_axiom ‹λ κ . «[Π]κ'κ»›
1713 by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
1714interpretation "russell-axiom[exe,2,2]": russel_axiom ‹λ κ . «[Π]κκ»›
1715 by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
1716interpretation "russell-axiom[exe,3,1,1]": russel_axiom ‹λ κ . «[Π]κκ'κ''»›
1717 by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1718interpretation "russell-axiom[exe,3,1,2]": russel_axiom ‹λ κ . «[Π]κ'κκ''»›
1719 by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1720interpretation "russell-axiom[exe,3,1,3]": russel_axiom ‹λ κ . «[Π]κ'κ''κ»›
1721 by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1722interpretation "russell-axiom[exe,3,2,1]": russel_axiom ‹λ κ . «[Π]κκκ'»›
1723 by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
1724interpretation "russell-axiom[exe,3,2,2]": russel_axiom ‹λ κ . «[Π]κκ'κ»›
1725 by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1726interpretation "russell-axiom[exe,3,2,3]": russel_axiom ‹λ κ . «[Π]κ'κκ»›
1727 by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1728interpretation "russell-axiom[exe,3,3]": russel_axiom ‹λ κ . «[Π]κκκ»›
1729 by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
1730
1731interpretation "russell-axiom[enc,1]": russel_axiom ‹λ κ . «κ[Π]»›
1732 by standard (metis "cqt:5:b[1]"[axiom_inst, THEN "→E"] "&E"(2))
1733interpretation "russell-axiom[enc,2,1]": russel_axiom ‹λ κ . «κκ'[Π]»›
1734 by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E")
1735interpretation "russell-axiom[enc,2,2]": russel_axiom ‹λ κ . «κ'κ[Π]»›
1736 by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
1737interpretation "russell-axiom[enc,2,3]": russel_axiom ‹λ κ . «κκ[Π]»›
1738 by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
1739interpretation "russell-axiom[enc,3,1,1]": russel_axiom ‹λ κ . «κκ'κ''[Π]»›
1740 by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1741interpretation "russell-axiom[enc,3,1,2]": russel_axiom ‹λ κ . «κ'κκ''[Π]»›
1742 by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1743interpretation "russell-axiom[enc,3,1,3]": russel_axiom ‹λ κ . «κ'κ''κ[Π]»›
1744 by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1745interpretation "russell-axiom[enc,3,2,1]": russel_axiom ‹λ κ . «κκκ'[Π]»›
1746 by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
1747interpretation "russell-axiom[enc,3,2,2]": russel_axiom ‹λ κ . «κκ'κ[Π]»›
1748 by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1749interpretation "russell-axiom[enc,3,2,3]": russel_axiom ‹λ κ . «κ'κκ[Π]»›
1750 by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1751interpretation "russell-axiom[enc,3,3]": russel_axiom ‹λ κ . «κκκ[Π]»›
1752 by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
1753
1754AOT_act_theorem "1-exists:1": ‹❙ιx φ{x}↓ ≡ ∃!x φ{x}›
1755proof(rule "≡I"; rule "→I")
1756 AOT_assume ‹❙ιx φ{x}↓›
1757 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1758 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1759 AOT_hence ‹φ{a} & ∀z (φ{z} → z = a)› using hintikka "≡E" by blast
1760 AOT_hence ‹∃x (φ{x} & ∀z (φ{z} → z = x))› by (rule "∃I")
1761 AOT_thus ‹∃!x φ{x}› using "uniqueness:1"[THEN "≡⇩d⇩fI"] by blast
1762next
1763 AOT_assume ‹∃!x φ{x}›
1764 AOT_hence ‹∃x (φ{x} & ∀z (φ{z} → z = x))›
1765 using "uniqueness:1"[THEN "≡⇩d⇩fE"] by blast
1766 then AOT_obtain b where ‹φ{b} & ∀z (φ{z} → z = b)› using "instantiation"[rotated] by blast
1767 AOT_hence ‹b = ❙ιx φ{x}› using hintikka "≡E" by blast
1768 AOT_thus ‹❙ιx φ{x}↓› by (metis "t=t-proper:2" "vdash-properties:6")
1769qed
1770
1771AOT_act_theorem "1-exists:2": ‹∃y(y=❙ιx φ{x}) ≡ ∃!x φ{x}›
1772 using "1-exists:1" "free-thms:1" "≡E"(6) by blast
1773
1774AOT_act_theorem "y-in:1": ‹x = ❙ιx φ{x} → φ{x}›
1775 using "&E"(1) "→I" hintikka "≡E"(1) by blast
1776
1777AOT_act_theorem "y-in:2": ‹z = ❙ιx φ{x} → φ{z}› using "y-in:1".
1778
1779AOT_act_theorem "y-in:3": ‹❙ιx φ{x}↓ → φ{❙ιx φ{x}}›
1780proof(rule "→I")
1781 AOT_assume ‹❙ιx φ{x}↓›
1782 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1783 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1784 moreover AOT_have ‹φ{a}› using calculation hintikka "≡E"(1) "&E" by blast
1785 ultimately AOT_show ‹φ{❙ιx φ{x}}› using "rule=E" by blast
1786qed
1787
1788AOT_act_theorem "y-in:4": ‹∃y (y = ❙ιx φ{x}) → φ{❙ιx φ{x}}›
1789 using "y-in:3"[THEN "→E"] "free-thms:1"[THEN "≡E"(2)] "→I" by blast
1790
1791
1792AOT_theorem "act-quant-nec": ‹∀β (❙𝒜φ{β} ≡ β = α) ≡ ∀β(❙𝒜❙𝒜φ{β} ≡ β = α)›
1793proof(rule "≡I"; rule "→I")
1794 AOT_assume ‹∀β (❙𝒜φ{β} ≡ β = α)›
1795 AOT_hence ‹❙𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1796 AOT_hence ‹❙𝒜❙𝒜φ{β} ≡ β = α› for β
1797 by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
1798 AOT_thus ‹∀β(❙𝒜❙𝒜φ{β} ≡ β = α)›
1799 by (rule "∀I")
1800next
1801 AOT_assume ‹∀β(❙𝒜❙𝒜φ{β} ≡ β = α)›
1802 AOT_hence ‹❙𝒜❙𝒜φ{β} ≡ β = α› for β using "∀E" by blast
1803 AOT_hence ‹❙𝒜φ{β} ≡ β = α› for β
1804 by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
1805 AOT_thus ‹∀β (❙𝒜φ{β} ≡ β = α)›
1806 by (rule "∀I")
1807qed
1808
1809AOT_theorem "equi-desc-descA:1": ‹x = ❙ιx φ{x} ≡ x = ❙ιx(❙𝒜φ{x})›
1810proof -
1811 AOT_have ‹x = ❙ιx φ{x} ≡ ∀z (❙𝒜φ{z} ≡ z = x)› using descriptions[axiom_inst] by blast
1812 also AOT_have ‹... ≡ ∀z (❙𝒜❙𝒜φ{z} ≡ z = x)›
1813 proof(rule "≡I"; rule "→I"; rule "∀I")
1814 AOT_assume ‹∀z (❙𝒜φ{z} ≡ z = x)›
1815 AOT_hence ‹❙𝒜φ{a} ≡ a = x› for a using "∀E" by blast
1816 AOT_thus ‹❙𝒜❙𝒜φ{a} ≡ a = x› for a by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
1817 next
1818 AOT_assume ‹∀z (❙𝒜❙𝒜φ{z} ≡ z = x)›
1819 AOT_hence ‹❙𝒜❙𝒜φ{a} ≡ a = x› for a using "∀E" by blast
1820 AOT_thus ‹❙𝒜φ{a} ≡ a = x› for a by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
1821 qed
1822 also AOT_have ‹... ≡ x = ❙ιx(❙𝒜φ{x})›
1823 using "Commutativity of ≡"[THEN "≡E"(1)] descriptions[axiom_inst] by fast
1824 finally show ?thesis .
1825qed
1826
1827AOT_theorem "equi-desc-descA:2": ‹❙ιx φ{x}↓ → ❙ιx φ{x} = ❙ιx(❙𝒜φ{x})›
1828proof(rule "→I")
1829 AOT_assume ‹❙ιx φ{x}↓›
1830 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1831 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1832 moreover AOT_have ‹a = ❙ιx(❙𝒜φ{x})› using calculation "equi-desc-descA:1"[THEN "≡E"(1)] by blast
1833 ultimately AOT_show ‹❙ιx φ{x} = ❙ιx(❙𝒜φ{x})› using "rule=E" by fast
1834qed
1835
1836AOT_theorem "nec-hintikka-scheme": ‹x = ❙ιx φ{x} ≡ ❙𝒜φ{x} & ∀z(❙𝒜φ{z} → z = x)›
1837proof -
1838 AOT_have ‹x = ❙ιx φ{x} ≡ ∀z(❙𝒜φ{z} ≡ z = x)› using descriptions[axiom_inst] by blast
1839 also AOT_have ‹… ≡ (❙𝒜φ{x} & ∀z(❙𝒜φ{z} → z = x))›
1840 using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" by fast
1841 finally show ?thesis.
1842qed
1843
1844AOT_theorem "equiv-desc-eq:1": ‹❙𝒜∀x(φ{x} ≡ ψ{x}) → ∀x (x = ❙ιx φ{x} ≡ x = ❙ιx ψ{x})›
1845proof(rule "→I"; rule "∀I")
1846 fix β
1847 AOT_assume ‹❙𝒜∀x(φ{x} ≡ ψ{x})›
1848 AOT_hence ‹❙𝒜(φ{x} ≡ ψ{x})› for x using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] "∀E"(2) by blast
1849 AOT_hence 0: ‹❙𝒜φ{x} ≡ ❙𝒜ψ{x}› for x by (metis "Act-Basic:5" "≡E"(1))
1850 AOT_have ‹β = ❙ιx φ{x} ≡ ❙𝒜φ{β} & ∀z(❙𝒜φ{z} → z = β)› using "nec-hintikka-scheme" by blast
1851 also AOT_have ‹... ≡ ❙𝒜ψ{β} & ∀z(❙𝒜ψ{z} → z = β)›
1852 proof (rule "≡I"; rule "→I")
1853 AOT_assume 1: ‹❙𝒜φ{β} & ∀z(❙𝒜φ{z} → z = β)›
1854 AOT_hence ‹❙𝒜φ{z} → z = β› for z using "&E" "∀E" by blast
1855 AOT_hence ‹❙𝒜ψ{z} → z = β› for z using 0 "≡E" "→I" "→E" by metis
1856 AOT_hence ‹∀z(❙𝒜ψ{z} → z = β)› using "∀I" by fast
1857 moreover AOT_have ‹❙𝒜ψ{β}› using "&E" 0[THEN "≡E"(1)] 1 by blast
1858 ultimately AOT_show ‹❙𝒜ψ{β} & ∀z(❙𝒜ψ{z} → z = β)› using "&I" by blast
1859 next
1860 AOT_assume 1: ‹❙𝒜ψ{β} & ∀z(❙𝒜ψ{z} → z = β)›
1861 AOT_hence ‹❙𝒜ψ{z} → z = β› for z using "&E" "∀E" by blast
1862 AOT_hence ‹❙𝒜φ{z} → z = β› for z using 0 "≡E" "→I" "→E" by metis
1863 AOT_hence ‹∀z(❙𝒜φ{z} → z = β)› using "∀I" by fast
1864 moreover AOT_have ‹❙𝒜φ{β}› using "&E" 0[THEN "≡E"(2)] 1 by blast
1865 ultimately AOT_show ‹❙𝒜φ{β} & ∀z(❙𝒜φ{z} → z = β)› using "&I" by blast
1866 qed
1867 also AOT_have ‹... ≡ β = ❙ιx ψ{x}›
1868 using "Commutativity of ≡"[THEN "≡E"(1)] "nec-hintikka-scheme" by blast
1869 finally AOT_show ‹β = ❙ιx φ{x} ≡ β = ❙ιx ψ{x}› .
1870qed
1871
1872AOT_theorem "equiv-desc-eq:2": ‹❙ιx φ{x}↓ & ❙𝒜∀x(φ{x} ≡ ψ{x}) → ❙ιx φ{x} = ❙ιx ψ{x}›
1873proof(rule "→I")
1874 AOT_assume ‹❙ιx φ{x}↓ & ❙𝒜∀x(φ{x} ≡ ψ{x})›
1875 AOT_hence 0: ‹∃y (y = ❙ιx φ{x})› and
1876 1: ‹∀x (x = ❙ιx φ{x} ≡ x = ❙ιx ψ{x})›
1877 using "&E" "free-thms:1"[THEN "≡E"(1)] "equiv-desc-eq:1" "→E" by blast+
1878 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1879 moreover AOT_have ‹a = ❙ιx ψ{x}› using calculation 1 "∀E" "≡E"(1) by fast
1880 ultimately AOT_show ‹❙ιx φ{x} = ❙ιx ψ{x}›
1881 using "rule=E" by fast
1882qed
1883
1884AOT_theorem "equiv-desc-eq:3": ‹❙ιx φ{x}↓ & □∀x(φ{x} ≡ ψ{x}) → ❙ιx φ{x} = ❙ιx ψ{x}›
1885 using "→I" "equiv-desc-eq:2"[THEN "→E", OF "&I"] "&E" "nec-imp-act"[THEN "→E"] by metis
1886
1887
1888AOT_theorem "equiv-desc-eq:4": ‹❙ιx φ{x}↓ → □❙ιx φ{x}↓›
1889proof(rule "→I")
1890 AOT_assume ‹❙ιx φ{x}↓›
1891 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1892 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1893 AOT_thus ‹□❙ιx φ{x}↓›
1894 using "ex:2:a" "rule=E" by fast
1895qed
1896
1897AOT_theorem "equiv-desc-eq:5": ‹❙ιx φ{x}↓ → ∃y □(y = ❙ιx φ{x})›
1898proof(rule "→I")
1899 AOT_assume ‹❙ιx φ{x}↓›
1900 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1901 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1902 AOT_hence ‹□(a = ❙ιx φ{x})› by (metis "id-nec:2" "vdash-properties:10")
1903 AOT_thus ‹∃y □(y = ❙ιx φ{x})› by (rule "∃I")
1904qed
1905
1906AOT_act_theorem "equiv-desc-eq2:1": ‹∀x (φ{x} ≡ ψ{x}) → ∀x (x = ❙ιx φ{x} ≡ x = ❙ιx ψ{x})›
1907 using "→I" "logic-actual"[act_axiom_inst, THEN "→E"] "equiv-desc-eq:1"[THEN "→E"]
1908 "RA[1]" "deduction-theorem" by blast
1909
1910AOT_act_theorem "equiv-desc-eq2:2": ‹❙ιx φ{x}↓ & ∀x (φ{x} ≡ ψ{x}) → ❙ιx φ{x} = ❙ιx ψ{x}›
1911 using "→I" "logic-actual"[act_axiom_inst, THEN "→E"] "equiv-desc-eq:2"[THEN "→E", OF "&I"]
1912 "RA[1]" "deduction-theorem" "&E" by metis
1913
1914context russel_axiom
1915begin
1916AOT_theorem "nec-russell-axiom": ‹ψ{❙ιx φ{x}} ≡ ∃x(❙𝒜φ{x} & ∀z(❙𝒜φ{z} → z = x) & ψ{x})›
1917proof -
1918 AOT_have b: ‹∀x (x = ❙ιx φ{x} ≡ (❙𝒜φ{x} & ∀z(❙𝒜φ{z} → z = x)))›
1919 using "nec-hintikka-scheme" "∀I" by fast
1920 show ?thesis
1921 proof(rule "≡I"; rule "→I")
1922 AOT_assume c: ‹ψ{❙ιx φ{x}}›
1923 AOT_hence d: ‹❙ιx φ{x}↓› using ψ_denotes_asm by blast
1924 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1925 then AOT_obtain a where a_def: ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1926 moreover AOT_have ‹a = ❙ιx φ{x} ≡ (❙𝒜φ{a} & ∀z(❙𝒜φ{z} → z = a))› using b "∀E" by blast
1927 ultimately AOT_have ‹❙𝒜φ{a} & ∀z(❙𝒜φ{z} → z = a)› using "≡E" by blast
1928 moreover AOT_have ‹ψ{a}›
1929 proof -
1930 AOT_have 1: ‹∀x∀y(x = y → y = x)›
1931 by (simp add: "id-eq:2" "universal-cor")
1932 AOT_have ‹a = ❙ιx φ{x} → ❙ιx φ{x} = a›
1933 by (rule "∀E"(1)[where τ="«❙ιx φ{x}»"]; rule "∀E"(2)[where β=a])
1934 (auto simp: d "universal-cor" 1)
1935 AOT_thus ‹ψ{a}›
1936 using a_def c "rule=E" "→E" by metis
1937 qed
1938 ultimately AOT_have ‹❙𝒜φ{a} & ∀z(❙𝒜φ{z} → z = a) & ψ{a}› by (rule "&I")
1939 AOT_thus ‹∃x(❙𝒜φ{x} & ∀z(❙𝒜φ{z} → z = x) & ψ{x})› by (rule "∃I")
1940 next
1941 AOT_assume ‹∃x(❙𝒜φ{x} & ∀z(❙𝒜φ{z} → z = x) & ψ{x})›
1942 then AOT_obtain b where g: ‹❙𝒜φ{b} & ∀z(❙𝒜φ{z} → z = b) & ψ{b}› using "instantiation"[rotated] by blast
1943 AOT_hence h: ‹b = ❙ιx φ{x} ≡ (❙𝒜φ{b} & ∀z(❙𝒜φ{z} → z = b))› using b "∀E" by blast
1944 AOT_have ‹❙𝒜φ{b} & ∀z(❙𝒜φ{z} → z = b)› and j: ‹ψ{b}› using g "&E" by blast+
1945 AOT_hence ‹b = ❙ιx φ{x}› using h "≡E" by blast
1946 AOT_thus ‹ψ{❙ιx φ{x}}› using j "rule=E" by blast
1947 qed
1948qed
1949end
1950
1951AOT_theorem "actual-desc:1": ‹❙ιx φ{x}↓ ≡ ∃!x ❙𝒜φ{x}›
1952proof (rule "≡I"; rule "→I")
1953 AOT_assume ‹❙ιx φ{x}↓›
1954 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1955 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1956 moreover AOT_have ‹a = ❙ιx φ{x} ≡ ∀z(❙𝒜φ{z} ≡ z = a)›
1957 using descriptions[axiom_inst] by blast
1958 ultimately AOT_have ‹∀z(❙𝒜φ{z} ≡ z = a)›
1959 using "≡E" by blast
1960 AOT_hence ‹∃x∀z(❙𝒜φ{z} ≡ z = x)› by (rule "∃I")
1961 AOT_thus ‹∃!x ❙𝒜φ{x}›
1962 using "uniqueness:2"[THEN "≡E"(2)] by fast
1963next
1964 AOT_assume ‹∃!x ❙𝒜φ{x}›
1965 AOT_hence ‹∃x∀z(❙𝒜φ{z} ≡ z = x)›
1966 using "uniqueness:2"[THEN "≡E"(1)] by fast
1967 then AOT_obtain a where ‹∀z(❙𝒜φ{z} ≡ z = a)› using "instantiation"[rotated] by blast
1968 moreover AOT_have ‹a = ❙ιx φ{x} ≡ ∀z(❙𝒜φ{z} ≡ z = a)›
1969 using descriptions[axiom_inst] by blast
1970 ultimately AOT_have ‹a = ❙ιx φ{x}› using "≡E" by blast
1971 AOT_thus ‹❙ιx φ{x}↓› by (metis "t=t-proper:2" "vdash-properties:6")
1972qed
1973
1974AOT_theorem "actual-desc:2": ‹x = ❙ιx φ{x} → ❙𝒜φ{x}›
1975 using "&E"(1) "contraposition:1[2]" "≡E"(1) "nec-hintikka-scheme" "reductio-aa:2" "vdash-properties:9" by blast
1976
1977AOT_theorem "actual-desc:3": ‹z = ❙ιx φ{x} → ❙𝒜φ{z}›
1978 using "actual-desc:2".
1979
1980AOT_theorem "actual-desc:4": ‹❙ιx φ{x}↓ → ❙𝒜φ{❙ιx φ{x}}›
1981proof(rule "→I")
1982 AOT_assume ‹❙ιx φ{x}↓›
1983 AOT_hence ‹∃y (y = ❙ιx φ{x})› by (metis "rule=I:1" "existential:1")
1984 then AOT_obtain a where ‹a = ❙ιx φ{x}› using "instantiation"[rotated] by blast
1985 AOT_thus ‹❙𝒜φ{❙ιx φ{x}}›
1986 using "actual-desc:2" "rule=E" "→E" by fast
1987qed
1988
1989
1990AOT_theorem "actual-desc:5": ‹❙ιx φ{x} = ❙ιx ψ{x} → ❙𝒜∀x(φ{x} ≡ ψ{x})›
1991proof(rule "→I")
1992 AOT_assume 0: ‹❙ιx φ{x} = ❙ιx ψ{x}›
1993 AOT_hence φ_down: ‹❙ιx φ{x}↓› and ψ_down: ‹❙ιx ψ{x}↓›
1994 using "t=t-proper:1" "t=t-proper:2" "vdash-properties:6" by blast+
1995 AOT_hence ‹∃y (y = ❙ιx φ{x})› and ‹∃y (y = ❙ιx ψ{x})› by (metis "rule=I:1" "existential:1")+
1996 then AOT_obtain a and b where a_eq: ‹a = ❙ιx φ{x}› and b_eq: ‹b = ❙ιx ψ{x}›
1997 using "instantiation"[rotated] by metis
1998
1999 AOT_have ‹∀α∀β (α = β → β = α)› by (rule "∀I"; rule "∀I"; rule "id-eq:2")
2000 AOT_hence ‹∀β (❙ιx φ{x} = β → β = ❙ιx φ{x})›
2001 using "∀E" φ_down by blast
2002 AOT_hence ‹❙ιx φ{x} = ❙ιx ψ{x} → ❙ιx ψ{x} = ❙ιx φ{x}›
2003 using "∀E" ψ_down by blast
2004 AOT_hence 1: ‹❙ιx ψ{x} = ❙ιx φ{x}› using 0
2005 "→E" by blast
2006
2007 AOT_have ‹❙𝒜φ{x} ≡ ❙𝒜ψ{x}› for x
2008 proof(rule "≡I"; rule "→I")
2009 AOT_assume ‹❙𝒜φ{x}›
2010 moreover AOT_have ‹❙𝒜φ{x} → x = a› for x
2011 using "nec-hintikka-scheme"[THEN "≡E"(1), OF a_eq, THEN "&E"(2)] "∀E" by blast
2012 ultimately AOT_have ‹x = a› using "→E" by blast
2013 AOT_hence ‹x = ❙ιx φ{x}› using a_eq "rule=E" by blast
2014 AOT_hence ‹x = ❙ιx ψ{x}› using 0 "rule=E" by blast
2015 AOT_thus ‹❙𝒜ψ{x}› by (metis "actual-desc:3" "vdash-properties:6")
2016 next
2017 AOT_assume ‹❙𝒜ψ{x}›
2018 moreover AOT_have ‹❙𝒜ψ{x} → x = b› for x
2019 using "nec-hintikka-scheme"[THEN "≡E"(1), OF b_eq, THEN "&E"(2)] "∀E" by blast
2020 ultimately AOT_have ‹x = b› using "→E" by blast
2021 AOT_hence ‹x = ❙ιx ψ{x}› using b_eq "rule=E" by blast
2022 AOT_hence ‹x = ❙ιx φ{x}› using 1 "rule=E" by blast
2023 AOT_thus ‹❙𝒜φ{x}› by (metis "actual-desc:3" "vdash-properties:6")
2024 qed
2025 AOT_hence ‹❙𝒜(φ{x} ≡ ψ{x})› for x by (metis "Act-Basic:5" "≡E"(2))
2026 AOT_hence ‹∀x ❙𝒜(φ{x} ≡ ψ{x})› by (rule "∀I")
2027 AOT_thus ‹❙𝒜∀x (φ{x} ≡ ψ{x})›
2028 using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
2029qed
2030
2031AOT_theorem "!box-desc:1": ‹∃!x □φ{x} → ∀y (y = ❙ιx φ{x} → φ{y})›
2032proof(rule "→I")
2033 AOT_assume ‹∃!x □φ{x}›
2034 AOT_hence ζ: ‹∃x (□φ{x} & ∀z (□φ{z} → z = x))›
2035 using "uniqueness:1"[THEN "≡⇩d⇩fE"] by blast
2036 then AOT_obtain b where θ: ‹□φ{b} & ∀z (□φ{z} → z = b)› using "instantiation"[rotated] by blast
2037 AOT_show ‹∀y (y = ❙ιx φ{x} → φ{y})›
2038 proof(rule GEN; rule "→I")
2039 fix y
2040 AOT_assume ‹y = ❙ιx φ{x}›
2041 AOT_hence ‹❙𝒜φ{y} & ∀z (❙𝒜φ{z} → z = y)› using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
2042 AOT_hence ‹❙𝒜φ{b} → b = y› using "&E" "∀E" by blast
2043 moreover AOT_have ‹❙𝒜φ{b}› using θ[THEN "&E"(1)] by (metis "nec-imp-act" "→E")
2044 ultimately AOT_have ‹b = y› using "→E" by blast
2045 moreover AOT_have ‹φ{b}› using θ[THEN "&E"(1)] by (metis "qml:2"[axiom_inst] "→E")
2046 ultimately AOT_show ‹φ{y}› using "rule=E" by blast
2047 qed
2048qed
2049
2050AOT_theorem "!box-desc:2": ‹∀x (φ{x} → □φ{x}) → (∃!x φ{x} → ∀y (y = ❙ιx φ{x} → φ{y}))›
2051proof(rule "→I"; rule "→I")
2052 AOT_assume ‹∀x (φ{x} → □φ{x})›
2053 moreover AOT_assume ‹∃!x φ{x}›
2054 ultimately AOT_have ‹∃!x □φ{x}›
2055 using "nec-exist-!"[THEN "→E", THEN "→E"] by blast
2056 AOT_thus ‹∀y (y = ❙ιx φ{x} → φ{y})›
2057 using "!box-desc:1" "→E" by blast
2058qed
2059
2060AOT_theorem "dr-alphabetic-thm": ‹❙ιν φ{ν}↓ → ❙ιν φ{ν} = ❙ιμ φ{μ}›
2061 by (simp add: "rule=I:1" "→I")
2062
2063AOT_theorem "RM:1[prem]": assumes ‹Γ ❙⊢⇩□ φ → ψ› shows ‹□Γ ❙⊢⇩□ □φ → □ψ›
2064proof -
2065 AOT_have ‹□Γ ❙⊢⇩□ □(φ → ψ)› using "RN[prem]" assms by blast
2066 AOT_thus ‹□Γ ❙⊢⇩□ □φ → □ψ› by (metis "qml:1"[axiom_inst] "→E")
2067qed
2068
2069AOT_theorem "RM:1": assumes ‹❙⊢⇩□ φ → ψ› shows ‹❙⊢⇩□ □φ → □ψ›
2070 using "RM:1[prem]" assms by blast
2071
2072lemmas RM = "RM:1"
2073
2074AOT_theorem "RM:2[prem]": assumes ‹Γ ❙⊢⇩□ φ → ψ› shows ‹□Γ ❙⊢⇩□ ◇φ → ◇ψ›
2075proof -
2076 AOT_have ‹Γ ❙⊢⇩□ ¬ψ → ¬φ› using assms
2077 by (simp add: "contraposition:1[1]")
2078 AOT_hence ‹□Γ ❙⊢⇩□ □¬ψ → □¬φ› using "RM:1[prem]" by blast
2079 AOT_thus ‹□Γ ❙⊢⇩□ ◇φ → ◇ψ›
2080 by (meson "≡⇩d⇩fE" "≡⇩d⇩fI" "conventions:5" "deduction-theorem" "modus-tollens:1")
2081qed
2082
2083AOT_theorem "RM:2": assumes ‹❙⊢⇩□ φ → ψ› shows ‹❙⊢⇩□ ◇φ → ◇ψ›
2084 using "RM:2[prem]" assms by blast
2085
2086lemmas "RM◇" = "RM:2"
2087
2088AOT_theorem "RM:3[prem]": assumes ‹Γ ❙⊢⇩□ φ ≡ ψ› shows ‹□Γ ❙⊢⇩□ □φ ≡ □ψ›
2089proof -
2090 AOT_have ‹Γ ❙⊢⇩□ φ → ψ› and ‹Γ ❙⊢⇩□ ψ → φ› using assms "≡E" "→I" by metis+
2091 AOT_hence ‹□Γ ❙⊢⇩□ □φ → □ψ› and ‹□Γ ❙⊢⇩□ □ψ → □φ› using "RM:1[prem]" by metis+
2092 AOT_thus ‹□Γ ❙⊢⇩□ □φ ≡ □ψ›
2093 by (simp add: "≡I")
2094qed
2095
2096AOT_theorem "RM:3": assumes ‹❙⊢⇩□ φ ≡ ψ› shows ‹❙⊢⇩□ □φ ≡ □ψ›
2097 using "RM:3[prem]" assms by blast
2098
2099lemmas RE = "RM:3"
2100
2101AOT_theorem "RM:4[prem]": assumes ‹Γ ❙⊢⇩□ φ ≡ ψ› shows ‹□Γ ❙⊢⇩□ ◇φ ≡ ◇ψ›
2102proof -
2103 AOT_have ‹Γ ❙⊢⇩□ φ → ψ› and ‹Γ ❙⊢⇩□ ψ → φ› using assms "≡E" "→I" by metis+
2104 AOT_hence ‹□Γ ❙⊢⇩□ ◇φ → ◇ψ› and ‹□Γ ❙⊢⇩□ ◇ψ → ◇φ› using "RM:2[prem]" by metis+
2105 AOT_thus ‹□Γ ❙⊢⇩□ ◇φ ≡ ◇ψ› by (simp add: "≡I")
2106qed
2107
2108AOT_theorem "RM:4": assumes ‹❙⊢⇩□ φ ≡ ψ› shows ‹❙⊢⇩□ ◇φ ≡ ◇ψ›
2109 using "RM:4[prem]" assms by blast
2110
2111lemmas "RE◇" = "RM:4"
2112
2113AOT_theorem "KBasic:1": ‹□φ → □(ψ → φ)›
2114 by (simp add: RM "pl:1"[axiom_inst])
2115
2116AOT_theorem "KBasic:2": ‹□¬φ → □(φ → ψ)›
2117 by (simp add: RM "useful-tautologies:3")
2118
2119AOT_theorem "KBasic:3": ‹□(φ & ψ) ≡ (□φ & □ψ)›
2120proof (rule "≡I"; rule "→I")
2121 AOT_assume ‹□(φ & ψ)›
2122 AOT_thus ‹□φ & □ψ›
2123 by (meson RM "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "vdash-properties:6")
2124next
2125 AOT_have ‹□φ → □(ψ → (φ & ψ))› by (simp add: "RM:1" Adjunction)
2126 AOT_hence ‹□φ → (□ψ → □(φ & ψ))› by (metis "Hypothetical Syllogism" "qml:1"[axiom_inst])
2127 moreover AOT_assume ‹□φ & □ψ›
2128 ultimately AOT_show ‹□(φ & ψ)›
2129 using "→E" "&E" by blast
2130qed
2131
2132AOT_theorem "KBasic:4": ‹□(φ ≡ ψ) ≡ (□(φ → ψ) & □(ψ → φ))›
2133proof -
2134 AOT_have θ: ‹□((φ → ψ) & (ψ → φ)) ≡ (□(φ → ψ) & □(ψ → φ))›
2135 by (fact "KBasic:3")
2136 AOT_modally_strict {
2137 AOT_have ‹(φ ≡ ψ) ≡ ((φ → ψ) & (ψ → φ))›
2138 by (fact "conventions:3"[THEN "≡Df"])
2139 }
2140 AOT_hence ξ: ‹□(φ ≡ ψ) ≡ □((φ → ψ) & (ψ → φ))›
2141 by (rule RE)
2142 with ξ and θ AOT_show ‹□(φ ≡ ψ) ≡ (□(φ → ψ) & □(ψ → φ))›
2143 using "≡E"(5) by blast
2144qed
2145
2146AOT_theorem "KBasic:5": ‹(□(φ → ψ) & □(ψ → φ)) → (□φ ≡ □ψ)›
2147proof -
2148 AOT_have ‹□(φ → ψ) → (□φ → □ψ)›
2149 by (fact "qml:1"[axiom_inst])
2150 moreover AOT_have ‹□(ψ → φ) → (□ψ → □φ)›
2151 by (fact "qml:1"[axiom_inst])
2152 ultimately AOT_have ‹(□(φ → ψ) & □(ψ → φ)) → ((□φ → □ψ) & (□ψ → □φ))›
2153 by (metis "&I" MP "Double Composition")
2154 moreover AOT_have ‹((□φ → □ψ) & (□ψ → □φ)) → (□φ ≡ □ψ)›
2155 using "conventions:3"[THEN "≡⇩d⇩fI"] "→I" by blast
2156 ultimately AOT_show ‹(□(φ → ψ) & □(ψ → φ)) → (□φ ≡ □ψ)›
2157 by (metis "Hypothetical Syllogism")
2158qed
2159
2160AOT_theorem "KBasic:6": ‹□(φ≡ ψ) → (□φ ≡ □ψ)›
2161 using "KBasic:4" "KBasic:5" "deduction-theorem" "≡E"(1) "vdash-properties:10" by blast
2162AOT_theorem "KBasic:7": ‹((□φ & □ψ) ∨ (□¬φ & □¬ψ)) → □(φ ≡ ψ)›
2163proof (rule "→I"; drule "∨E"(1); (rule "→I")?)
2164 AOT_assume ‹□φ & □ψ›
2165 AOT_hence ‹□φ› and ‹□ψ› using "&E" by blast+
2166 AOT_hence ‹□(φ → ψ)› and ‹□(ψ → φ)› using "KBasic:1" "→E" by blast+
2167 AOT_hence ‹□(φ → ψ) & □(ψ → φ)› using "&I" by blast
2168 AOT_thus ‹□(φ ≡ ψ)› by (metis "KBasic:4" "≡E"(2))
2169next
2170 AOT_assume ‹□¬φ & □¬ψ›
2171 AOT_hence 0: ‹□(¬φ & ¬ψ)› using "KBasic:3"[THEN "≡E"(2)] by blast
2172 AOT_modally_strict {
2173 AOT_have ‹(¬φ & ¬ψ) → (φ ≡ ψ)›
2174 by (metis "&E"(1) "&E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
2175 }
2176 AOT_hence ‹□(¬φ & ¬ψ) → □(φ ≡ ψ)›
2177 by (rule RM)
2178 AOT_thus ‹□(φ ≡ ψ)› using 0 "→E" by blast
2179qed(auto)
2180
2181AOT_theorem "KBasic:8": ‹□(φ & ψ) → □(φ ≡ ψ)›
2182 by (meson "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I")
2183AOT_theorem "KBasic:9": ‹□(¬φ & ¬ψ) → □(φ ≡ ψ)›
2184 by (metis "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I" "raa-cor:4")
2185AOT_theorem "KBasic:10": ‹□φ ≡ □¬¬φ›
2186 by (simp add: "RM:3" "oth-class-taut:3:b")
2187AOT_theorem "KBasic:11": ‹¬□φ ≡ ◇¬φ›
2188proof (rule "≡I"; rule "→I")
2189 AOT_show ‹◇¬φ› if ‹¬□φ›
2190 using that "≡⇩d⇩fI" "conventions:5" "KBasic:10" "≡E"(3) by blast
2191next
2192 AOT_show ‹¬□φ› if ‹◇¬φ›
2193 using "≡⇩d⇩fE" "conventions:5" "KBasic:10" "≡E"(4) that by blast
2194qed
2195AOT_theorem "KBasic:12": ‹□φ ≡ ¬◇¬φ›
2196proof (rule "≡I"; rule "→I")
2197 AOT_show ‹¬◇¬φ› if ‹□φ›
2198 using "¬¬I" "KBasic:11" "≡E"(3) that by blast
2199next
2200 AOT_show ‹□φ› if ‹¬◇¬φ›
2201 using "KBasic:11" "≡E"(1) "reductio-aa:1" that by blast
2202qed
2203AOT_theorem "KBasic:13": ‹□(φ → ψ) → (◇φ → ◇ψ)›
2204proof -
2205 AOT_have ‹φ → ψ ❙⊢⇩□ φ → ψ› by blast
2206 AOT_hence ‹□(φ → ψ) ❙⊢⇩□ ◇φ → ◇ψ›
2207 using "RM:2[prem]" by blast
2208 AOT_thus ‹□(φ → ψ) → (◇φ → ◇ψ)› using "→I" by blast
2209qed
2210lemmas "K◇" = "KBasic:13"
2211AOT_theorem "KBasic:14": ‹◇□φ ≡ ¬□◇¬φ›
2212 by (meson "RE◇" "KBasic:11" "KBasic:12" "≡E"(6) "oth-class-taut:3:a")
2213AOT_theorem "KBasic:15": ‹(□φ ∨ □ψ) → □(φ ∨ ψ)›
2214proof -
2215 AOT_modally_strict {
2216 AOT_have ‹φ → (φ ∨ ψ)› and ‹ψ → (φ ∨ ψ)›
2217 by (auto simp: "Disjunction Addition"(1) "Disjunction Addition"(2))
2218 }
2219 AOT_hence ‹□φ → □(φ ∨ ψ)› and ‹□ψ → □(φ ∨ ψ)›
2220 using RM by blast+
2221 AOT_thus ‹(□φ ∨ □ψ) → □(φ ∨ ψ)›
2222 by (metis "∨E"(1) "deduction-theorem")
2223qed
2224
2225AOT_theorem "KBasic:16": ‹(□φ & ◇ψ) → ◇(φ & ψ)›
2226 by (meson "KBasic:13" "RM:1" Adjunction "Hypothetical Syllogism" Importation "vdash-properties:6")
2227
2228AOT_theorem "rule-sub-lem:1:a":
2229 assumes ‹❙⊢⇩□ □(ψ ≡ χ)›
2230 shows ‹❙⊢⇩□ ¬ψ ≡ ¬χ›
2231 using "qml:2"[axiom_inst, THEN "→E", OF assms]
2232 "≡E"(1) "oth-class-taut:4:b" by blast
2233
2234AOT_theorem "rule-sub-lem:1:b":
2235 assumes ‹❙⊢⇩□ □(ψ ≡ χ)›
2236 shows ‹❙⊢⇩□ (ψ → Θ) ≡ (χ → Θ)›
2237 using "qml:2"[axiom_inst, THEN "→E", OF assms]
2238 using "oth-class-taut:4:c" "vdash-properties:6" by blast
2239
2240AOT_theorem "rule-sub-lem:1:c":
2241 assumes ‹❙⊢⇩□ □(ψ ≡ χ)›
2242 shows ‹❙⊢⇩□ (Θ → ψ) ≡ (Θ → χ)›
2243 using "qml:2"[axiom_inst, THEN "→E", OF assms]
2244 using "oth-class-taut:4:d" "vdash-properties:6" by blast
2245
2246AOT_theorem "rule-sub-lem:1:d":
2247 assumes ‹for arbitrary α: ❙⊢⇩□ □(ψ{α} ≡ χ{α})›
2248 shows ‹❙⊢⇩□ ∀α ψ{α} ≡ ∀α χ{α}›
2249proof -
2250 AOT_modally_strict {
2251 AOT_have ‹∀α (ψ{α} ≡ χ{α})›
2252 using "qml:2"[axiom_inst, THEN "→E", OF assms] "∀I" by fast
2253 AOT_hence 0: ‹ψ{α} ≡ χ{α}› for α using "∀E" by blast
2254 AOT_show ‹∀α ψ{α} ≡ ∀α χ{α}›
2255 proof (rule "≡I"; rule "→I")
2256 AOT_assume ‹∀α ψ{α}›
2257 AOT_hence ‹ψ{α}› for α using "∀E" by blast
2258 AOT_hence ‹χ{α}› for α using 0 "≡E" by blast
2259 AOT_thus ‹∀α χ{α}› by (rule "∀I")
2260 next
2261 AOT_assume ‹∀α χ{α}›
2262 AOT_hence ‹χ{α}› for α using "∀E" by blast
2263 AOT_hence ‹ψ{α}› for α using 0 "≡E" by blast
2264 AOT_thus ‹∀α ψ{α}› by (rule "∀I")
2265 qed
2266 }
2267qed
2268
2269AOT_theorem "rule-sub-lem:1:e":
2270 assumes ‹❙⊢⇩□ □(ψ ≡ χ)›
2271 shows ‹❙⊢⇩□ [λ ψ] ≡ [λ χ]›
2272 using "qml:2"[axiom_inst, THEN "→E", OF assms]
2273 using "≡E"(1) "propositions-lemma:6" by blast
2274
2275AOT_theorem "rule-sub-lem:1:f":
2276 assumes ‹❙⊢⇩□ □(ψ ≡ χ)›
2277 shows ‹❙⊢⇩□ ❙𝒜ψ ≡ ❙𝒜χ›
2278 using "qml:2"[axiom_inst, THEN "→E", OF assms, THEN "RA[2]"]
2279 by (metis "Act-Basic:5" "≡E"(1))
2280
2281AOT_theorem "rule-sub-lem:1:g":
2282 assumes ‹❙⊢⇩□ □(ψ ≡ χ)›
2283 shows ‹❙⊢⇩□ □ψ ≡ □χ›
2284 using "KBasic:6" assms "vdash-properties:6" by blast
2285
2286text‹Note that instead of deriving @{text "rule-sub-lem:2"}, @{text "rule-sub-lem:3"}, @{text "rule-sub-lem:4"},
2287 and @{text "rule-sub-nec"}, we construct substitution methods instead.›
2288
2289class AOT_subst =
2290 fixes AOT_subst :: "('a ⇒ 𝗈) ⇒ bool"
2291 and AOT_subst_cond :: "'a ⇒ 'a ⇒ bool"
2292 assumes AOT_subst: "AOT_subst φ ⟹ AOT_subst_cond ψ χ ⟹ [v ⊨ «φ ψ» ≡ «φ χ»]"
2293
2294named_theorems AOT_substI
2295
2296instantiation 𝗈 :: AOT_subst
2297begin
2298
2299inductive AOT_subst_𝗈 where
2300 AOT_subst_𝗈_id[AOT_substI]: "AOT_subst_𝗈 (λφ. φ)"
2301| AOT_subst_𝗈_const[AOT_substI]: "AOT_subst_𝗈 (λφ. ψ)"
2302| AOT_subst_𝗈_not[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «¬Θ{φ}»)"
2303| AOT_subst_𝗈_imp[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 Ξ ⟹ AOT_subst_𝗈 (λ φ. «Θ{φ} → Ξ{φ}»)"
2304| AOT_subst_𝗈_lambda0[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. (AOT_lambda0 (Θ φ)))"
2305| AOT_subst_𝗈_act[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «❙𝒜Θ{φ}»)"
2306| AOT_subst_𝗈_box[AOT_substI]: "AOT_subst_𝗈 Θ ⟹ AOT_subst_𝗈 (λ φ. «□Θ{φ}»)"
2307| AOT_subst_𝗈_by_def[AOT_substI]: "(⋀ ψ . AOT_model_equiv_def (Θ ψ) (Ξ ψ)) ⟹ AOT_subst_𝗈 Ξ ⟹ AOT_subst_𝗈 Θ"
2308
2309definition AOT_subst_cond_𝗈 where "AOT_subst_cond_𝗈 ≡ λ ψ χ . ∀ v . [v ⊨ ψ ≡ χ]"
2310
2311instance
2312proof
2313 fix ψ χ :: 𝗈 and φ :: ‹𝗈 ⇒ 𝗈›
2314 assume cond: ‹AOT_subst_cond ψ χ›
2315 assume ‹AOT_subst φ›
2316 moreover AOT_have ‹❙⊢⇩□ ψ ≡ χ› using cond unfolding AOT_subst_cond_𝗈_def by blast
2317 ultimately AOT_show ‹❙⊢⇩□ φ{ψ} ≡ φ{χ}›
2318 proof (induct arbitrary: ψ χ)
2319 case AOT_subst_𝗈_id
2320 thus ?case using "≡E"(2) "oth-class-taut:4:b" "rule-sub-lem:1:a" by blast
2321 next
2322 case (AOT_subst_𝗈_const ψ)
2323 thus ?case by (simp add: "oth-class-taut:3:a")
2324 next
2325 case (AOT_subst_𝗈_not Θ)
2326 thus ?case by (simp add: RN "rule-sub-lem:1:a")
2327 next
2328 case (AOT_subst_𝗈_imp Θ Ξ)
2329 thus ?case by (meson RN "≡E"(5) "rule-sub-lem:1:b" "rule-sub-lem:1:c")
2330 next
2331 case (AOT_subst_𝗈_lambda0 Θ)
2332 thus ?case by (simp add: RN "rule-sub-lem:1:e")
2333 next
2334 case (AOT_subst_𝗈_act Θ)
2335 thus ?case by (simp add: RN "rule-sub-lem:1:f")
2336 next
2337 case (AOT_subst_𝗈_box Θ)
2338 thus ?case by (simp add: RN "rule-sub-lem:1:g")
2339 next
2340 case (AOT_subst_𝗈_by_def Θ Ξ)
2341 AOT_modally_strict {
2342 AOT_have ‹Ξ{ψ} ≡ Ξ{χ}› using AOT_subst_𝗈_by_def by simp
2343 AOT_thus ‹Θ{ψ} ≡ Θ{χ}›
2344 using "≡Df"[OF AOT_subst_𝗈_by_def(1), of _ ψ] "≡Df"[OF AOT_subst_𝗈_by_def(1), of _ χ]
2345 by (metis "≡E"(6) "oth-class-taut:3:a")
2346 }
2347 qed
2348qed
2349end
2350
2351instantiation "fun" :: (AOT_Term_id_2, AOT_subst) AOT_subst
2352begin
2353
2354definition AOT_subst_cond_fun :: "('a ⇒ 'b) ⇒ ('a ⇒ 'b) ⇒ bool" where
2355 "AOT_subst_cond_fun ≡ λ φ ψ . ∀ α . AOT_subst_cond (φ (AOT_term_of_var α)) (ψ (AOT_term_of_var α))"
2356
2357inductive AOT_subst_fun :: "(('a ⇒ 'b) ⇒ 𝗈) ⇒ bool" where
2358 AOT_subst_fun_const[AOT_substI]: "AOT_subst_fun (λφ. ψ)"
2359| AOT_subst_fun_id[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst_fun (λφ. Ψ (φ (AOT_term_of_var x)))"
2360| AOT_subst_fun_all[AOT_substI]: "AOT_subst Ψ ⟹ (⋀ α . AOT_subst_fun (Θ (AOT_term_of_var α))) ⟹ AOT_subst_fun (λφ :: 'a ⇒ 'b. Ψ «∀α «Θ (α::'a) φ»»)"
2361| AOT_subst_fun_not[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst_fun (λφ. «¬«Ψ φ»»)"
2362| AOT_subst_fun_imp[AOT_substI]: "AOT_subst Ψ ⟹ AOT_subst Θ ⟹ AOT_subst_fun (λφ. ««Ψ φ» → «Θ φ»»)"
2363| AOT_subst_fun_lambda0[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. (AOT_lambda0 (Θ φ)))"
2364| AOT_subst_fun_act[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. «❙𝒜«Θ φ»»)"
2365| AOT_subst_fun_box[AOT_substI]: "AOT_subst Θ ⟹ AOT_subst_fun (λ φ. «□«Θ φ»»)"
2366| AOT_subst_fun_def[AOT_substI]: "(⋀ φ . AOT_model_equiv_def (Θ φ) (Ψ φ)) ⟹ AOT_subst_fun Ψ ⟹ AOT_subst_fun Θ"
2367
2368instance proof
2369 fix ψ χ :: "'a ⇒ 'b" and φ :: ‹('a ⇒ 'b) ⇒ 𝗈›
2370 assume ‹AOT_subst φ›
2371 moreover assume cond: ‹AOT_subst_cond ψ χ›
2372 ultimately AOT_show ‹❙⊢⇩□ «φ ψ» ≡ «φ χ»›
2373 proof(induct)
2374 case (AOT_subst_fun_const ψ)
2375 then show ?case by (simp add: "oth-class-taut:3:a")
2376 next
2377 case (AOT_subst_fun_id Ψ x)
2378 then show ?case by (simp add: AOT_subst AOT_subst_cond_fun_def)
2379 next
2380 case (AOT_subst_fun_all Ψ Θ)
2381 AOT_have ‹❙⊢⇩□ □(Θ{α, «ψ»} ≡ Θ{α, «χ»})› for α
2382 using AOT_subst_fun_all.hyps(3) AOT_subst_fun_all.prems RN by presburger
2383 thus ?case using AOT_subst[OF AOT_subst_fun_all(1)]
2384 by (simp add: RN "rule-sub-lem:1:d" AOT_subst_cond_fun_def AOT_subst_cond_𝗈_def)
2385 next
2386 case (AOT_subst_fun_not Ψ)
2387 then show ?case by (simp add: RN "rule-sub-lem:1:a")
2388 next
2389 case (AOT_subst_fun_imp Ψ Θ)
2390 then show ?case
2391 unfolding AOT_subst_cond_fun_def AOT_subst_cond_𝗈_def
2392 by (meson "≡E"(5) "oth-class-taut:4:c" "oth-class-taut:4:d" "vdash-properties:6")
2393 next
2394 case (AOT_subst_fun_lambda0 Θ)
2395 then show ?case by (simp add: RN "rule-sub-lem:1:e")
2396 next
2397 case (AOT_subst_fun_act Θ)
2398 then show ?case by (simp add: RN "rule-sub-lem:1:f")
2399 next
2400 case (AOT_subst_fun_box Θ)
2401 then show ?case by (simp add: RN "rule-sub-lem:1:g")
2402 next
2403 case (AOT_subst_fun_def Θ Ψ)
2404 then show ?case
2405 by (meson "df-rules-formulas[3]" "df-rules-formulas[4]" "≡I" "≡E"(5))
2406 qed
2407qed
2408end
2409
2410method_setup AOT_defI =
2411‹Scan.lift (Scan.succeed (fn ctxt => (Method.CONTEXT_METHOD (fn thms => (Context_Tactic.CONTEXT_SUBGOAL (fn (trm,int) =>
2412Context_Tactic.CONTEXT_TACTIC (
2413let
2414fun findHeadConst (Const x) = SOME x
2415 | findHeadConst (A $ B) = findHeadConst A
2416 | findHeadConst _ = NONE
2417fun findDef (Const (\<^const_name>‹AOT_model_equiv_def›, _) $ lhs $ rhs) = findHeadConst lhs
2418 | findDef (A $ B) = (case findDef A of SOME x => SOME x | _ => findDef B)
2419 | findDef (Abs (a,b,c)) = findDef c
2420 | findDef _ = NONE
2421val const_opt = (findDef trm)
2422val defs = case const_opt of SOME const => List.filter (fn thm => let
2423 val concl = Thm.concl_of thm
2424 val thmconst = (findDef concl)
2425 in case thmconst of SOME (c,_) => fst const = c | _ => false end) (AOT_Definitions.get ctxt)
2426 | _ => []
2427in
2428resolve_tac ctxt defs 1
2429end
2430)) 1)))))›
2431‹Resolve AOT definitions›
2432
2433
2434method AOT_subst_intro_helper = ((rule AOT_substI
2435 | AOT_defI
2436 | (simp only: AOT_subst_cond_𝗈_def AOT_subst_cond_fun_def; ((rule allI)+)?)))
2437
2438method_setup AOT_subst = ‹
2439Scan.option (Scan.lift (Args.parens (Args.$$$ "reverse"))) --
2440Scan.lift (Args.embedded_inner_syntax -- Args.embedded_inner_syntax
2441) -- Scan.option (Scan.lift (Args.$$$ "bound" -- Args.colon) |-- Scan.repeat1 (Scan.lift (Args.embedded_inner_syntax)))
2442>> (fn ((reversed,(raw_p,raw_q)),raw_bounds) => (fn ctxt =>
2443(Method.SIMPLE_METHOD (Subgoal.FOCUS (fn {context = ctxt, params = _, prems = prems, asms = asms, concl = concl, schematics = _} =>
2444let
2445val thms = prems
2446val ctxt' = ctxt
2447val ctxt = Context_Position.set_visible false ctxt
2448
2449val p = AOT_read_term @{nonterminal φ'} ctxt raw_p
2450val p = Syntax.check_term ctxt p
2451val ctxt = Variable.declare_term p ctxt
2452val q = AOT_read_term @{nonterminal φ'} ctxt raw_q
2453val q = Syntax.check_term ctxt q
2454val ctxt = Variable.declare_term q ctxt
2455
2456
2457val raw_bounds = case raw_bounds of SOME bounds => bounds | _ => []
2458val bounds = (map (fn x => Syntax.check_term ctxt (AOT_read_term @{nonterminal τ'} ctxt x))) raw_bounds
2459val p = fold (fn bound => fn p => let in Term.abs ("α", Term.type_of bound) (Term.abstract_over (bound,p)) end) bounds p
2460val p = Syntax.check_term ctxt p
2461val p_ty = Term.type_of p
2462
2463val pat = @{const Trueprop} $ (@{const AOT_model_valid_in} $ Var (("w",0), @{typ w}) $ (Var (("φ",0), Type (\<^type_name>‹fun›, [p_ty, @{typ 𝗈}])) $ p))
2464val univ = Unify.matchers (Context.Proof ctxt) [(pat, Thm.term_of concl)]
2465val univ = hd (Seq.list_of univ)
2466val phi = the (Envir.lookup univ (("φ",0), Type (\<^type_name>‹fun›, [p_ty, @{typ 𝗈}])))
2467
2468val q = fold (fn bound => fn q => let in Term.abs ("α", Term.type_of bound) (Term.abstract_over (bound,q)) end) bounds q
2469val q = Syntax.check_term ctxt q
2470
2471
2472val ctxt = Context_Position.restore_visible ctxt' ctxt
2473val ctxt' = ctxt
2474fun unsource str = fst (Input.source_content (Syntax.read_input str))
2475val (_,ctxt') = Proof_Context.add_fixes (map (fn str => (Binding.make (unsource str, Position.none), NONE, Mixfix.NoSyn)) raw_bounds) ctxt'
2476val _ = (map (fn x => Syntax.check_term ctxt (AOT_read_term @{nonterminal τ'} ctxt' x))) raw_bounds
2477val _ = AOT_read_term @{nonterminal φ'} ctxt' raw_p
2478val _ = AOT_read_term @{nonterminal φ'} ctxt' raw_q
2479
2480val abs = phi
2481val abs = HOLogic.mk_Trueprop (@{const AOT_subst(_)} $ abs)
2482val abs = Syntax.check_term ctxt abs
2483
2484val substThm = Goal.prove ctxt [] [] abs (fn {context=ctxt, prems=prems} =>
2485 REPEAT (SUBGOAL (fn (trm,int) => let
2486 fun findHeadConst (Const x) = SOME x
2487 | findHeadConst (A $ B) = findHeadConst A
2488 | findHeadConst _ = NONE
2489 fun findDef (Const (\<^const_name>‹AOT_model_equiv_def›, _) $ lhs $ rhs) = findHeadConst lhs
2490 | findDef (A $ B) = (case findDef A of SOME x => SOME x | _ => findDef B)
2491 | findDef (Abs (a,b,c)) = findDef c
2492 | findDef _ = NONE
2493 val const_opt = (findDef trm)
2494 val defs = case const_opt of SOME const => List.filter (fn thm => let
2495 val concl = Thm.concl_of thm
2496 val thmconst = (findDef concl)
2497 in case thmconst of SOME (c,_) => fst const = c | _ => false end) (AOT_Definitions.get ctxt)
2498 | _ => []
2499 val tac = case defs of [] => safe_step_tac (ctxt addSIs @{thms AOT_substI}) 1
2500 | _ => resolve_tac ctxt defs 1
2501 in tac end) 1)
2502 )
2503val substThm = substThm RS @{thm AOT_subst}
2504val abs = Thm.cterm_of ctxt abs
2505val substThm = case reversed of NONE =>
2506 let
2507 val substThm = Drule.instantiate_normalize ([],[((("ψ", 0), p_ty), Thm.cterm_of ctxt p),
2508 ((("χ", 0), p_ty), Thm.cterm_of ctxt q)]) substThm
2509 val substThm = substThm RS @{thm "≡E"(2)}
2510 in substThm end
2511| _ => let
2512 val substThm = Drule.instantiate_normalize ([],[((("χ", 0), p_ty), Thm.cterm_of ctxt p),
2513 ((("ψ", 0), p_ty), Thm.cterm_of ctxt q)]) substThm
2514 val substThm = substThm RS @{thm "≡E"(1)}
2515 in substThm end
2516
2517in
2518resolve_tac ctxt [substThm] 1
2519THEN simp_tac (ctxt addsimps [@{thm AOT_subst_cond_𝗈_def}, @{thm AOT_subst_cond_fun_def}]) 1
2520THEN (REPEAT (resolve_tac ctxt [@{thm allI}] 1))
2521THEN (TRY (resolve_tac ctxt thms 1))
2522end
2523) ctxt 1))))
2524›
2525
2526method AOT_subst_old for ψ::"'a::AOT_subst" and χ::"'a::AOT_subst" =
2527 (match conclusion in "[v ⊨ «φ ψ»]" for φ and v ⇒
2528 ‹match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail›
2529 ¦ _ ⇒ ‹rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(2)]
2530 ; (AOT_subst_intro_helper+)?››)
2531
2532method AOT_subst_old_rev for χ::"'a::AOT_subst" and ψ::"'a::AOT_subst" =
2533 (match conclusion in "[v ⊨ «φ ψ»]" for φ and v ⇒
2534 ‹match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail›
2535 ¦ _ ⇒ ‹rule AOT_subst[where φ=φ and ψ=χ and χ=ψ, THEN "≡E"(1)]
2536 ; (AOT_subst_intro_helper+)?››)
2537
2538method AOT_subst_manual for φ::"'a::AOT_subst ⇒ 𝗈" =
2539 (rule AOT_subst[where φ=φ, THEN "≡E"(2)]; (AOT_subst_intro_helper+)?)
2540
2541method AOT_subst_manual_rev for φ::"'a::AOT_subst ⇒ 𝗈" =
2542 (rule AOT_subst[where φ=φ, THEN "≡E"(1)]; (AOT_subst_intro_helper+)?)
2543
2544method AOT_subst_using uses subst =
2545 (match subst in "[?w ⊨ ψ ≡ χ]" for ψ χ ⇒ ‹
2546 match conclusion in "[v ⊨ «φ ψ»]" for φ v ⇒ ‹
2547 rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(2)]
2548 ; ((AOT_subst_intro_helper | (fact subst; fail))+)?››)
2549
2550method AOT_subst_using_rev uses subst =
2551 (match subst in "[?w ⊨ ψ ≡ χ]" for ψ χ ⇒ ‹
2552 match conclusion in "[v ⊨ «φ χ»]" for φ v ⇒ ‹
2553 rule AOT_subst[where φ=φ and ψ=ψ and χ=χ, THEN "≡E"(1)]
2554 ; ((AOT_subst_intro_helper | (fact subst; fail))+)?››)
2555
2556AOT_theorem "rule-sub-remark:1[1]": assumes ‹❙⊢⇩□ A!x ≡ ¬◇E!x› and ‹¬A!x› shows ‹¬¬◇E!x›
2557 by (AOT_subst (reverse) ‹¬◇E!x› ‹A!x›)
2558 (auto simp: assms)
2559
2560AOT_theorem "rule-sub-remark:1[2]": assumes ‹❙⊢⇩□ A!x ≡ ¬◇E!x› and ‹¬¬◇E!x› shows ‹¬A!x›
2561 by (AOT_subst ‹A!x› ‹¬◇E!x›)
2562 (auto simp: assms)
2563
2564AOT_theorem "rule-sub-remark:2[1]":
2565 assumes ‹❙⊢⇩□ [R]xy ≡ ([R]xy & ([Q]a ∨ ¬[Q]a))› and ‹p → [R]xy› shows ‹p → [R]xy & ([Q]a ∨ ¬[Q]a)›
2566 by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2567
2568AOT_theorem "rule-sub-remark:2[2]":
2569 assumes ‹❙⊢⇩□ [R]xy ≡ ([R]xy & ([Q]a ∨ ¬[Q]a))› and ‹p → [R]xy & ([Q]a ∨ ¬[Q]a)› shows ‹p → [R]xy›
2570 by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2571
2572AOT_theorem "rule-sub-remark:3[1]":
2573 assumes ‹for arbitrary x: ❙⊢⇩□ A!x ≡ ¬◇E!x›
2574 and ‹∃x A!x›
2575 shows ‹∃x ¬◇E!x›
2576 by (AOT_subst (reverse) ‹¬◇E!x› ‹A!x› bound: x)
2577 (auto simp: assms)
2578
2579AOT_theorem "rule-sub-remark:3[2]":
2580 assumes ‹for arbitrary x: ❙⊢⇩□ A!x ≡ ¬◇E!x›
2581 and ‹∃x ¬◇E!x›
2582 shows ‹∃x A!x›
2583 by (AOT_subst ‹A!x› ‹¬◇E!x› bound: x)
2584 (auto simp: assms)
2585
2586AOT_theorem "rule-sub-remark:4[1]":
2587 assumes ‹❙⊢⇩□ ¬¬[P]x ≡ [P]x› and ‹❙𝒜¬¬[P]x› shows ‹❙𝒜[P]x›
2588 by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2589
2590AOT_theorem "rule-sub-remark:4[2]":
2591 assumes ‹❙⊢⇩□ ¬¬[P]x ≡ [P]x› and ‹❙𝒜[P]x› shows ‹❙𝒜¬¬[P]x›
2592 by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2593
2594AOT_theorem "rule-sub-remark:5[1]":
2595 assumes ‹❙⊢⇩□ (φ → ψ) ≡ (¬ψ → ¬φ)› and ‹□(φ → ψ)› shows ‹□(¬ψ → ¬φ)›
2596 by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2597
2598AOT_theorem "rule-sub-remark:5[2]":
2599 assumes ‹❙⊢⇩□ (φ → ψ) ≡ (¬ψ → ¬φ)› and ‹□(¬ψ → ¬φ)› shows ‹□(φ → ψ)›
2600 by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2601
2602AOT_theorem "rule-sub-remark:6[1]":
2603 assumes ‹❙⊢⇩□ ψ ≡ χ› and ‹□(φ → ψ)› shows ‹□(φ → χ)›
2604 by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2605
2606AOT_theorem "rule-sub-remark:6[2]":
2607 assumes ‹❙⊢⇩□ ψ ≡ χ› and ‹□(φ → χ)› shows ‹□(φ → ψ)›
2608 by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2609
2610AOT_theorem "rule-sub-remark:7[1]":
2611 assumes ‹❙⊢⇩□ φ ≡ ¬¬φ› and ‹□(φ → φ)› shows ‹□(¬¬φ → φ)›
2612 by (AOT_subst_using_rev subst: assms(1)) (simp add: assms(2))
2613
2614AOT_theorem "rule-sub-remark:7[2]":
2615 assumes ‹❙⊢⇩□ φ ≡ ¬¬φ› and ‹□(¬¬φ → φ)› shows ‹□(φ → φ)›
2616 by (AOT_subst_using subst: assms(1)) (simp add: assms(2))
2617
2618AOT_theorem "KBasic2:1": ‹□¬φ ≡ ¬◇φ›
2619 by (meson "conventions:5" "contraposition:2" "Hypothetical Syllogism" "df-rules-formulas[3]"
2620 "df-rules-formulas[4]" "≡I" "useful-tautologies:1")
2621
2622AOT_theorem "KBasic2:2": ‹◇(φ ∨ ψ) ≡ (◇φ ∨ ◇ψ)›
2623proof -
2624 AOT_have ‹◇(φ ∨ ψ) ≡ ◇¬(¬φ & ¬ψ)›
2625 by (simp add: "RE◇" "oth-class-taut:5:b")
2626 also AOT_have ‹… ≡ ¬□(¬φ & ¬ψ)›
2627 using "KBasic:11" "≡E"(6) "oth-class-taut:3:a" by blast
2628 also AOT_have ‹… ≡ ¬(□¬φ & □¬ψ)›
2629 using "KBasic:3" "≡E"(1) "oth-class-taut:4:b" by blast
2630 also AOT_have ‹… ≡ ¬(¬◇φ & ¬◇ψ)›
2631 using "KBasic2:1"
2632 by (AOT_subst ‹□¬φ› ‹¬◇φ›; AOT_subst ‹□¬ψ› ‹¬◇ψ›; auto simp: "oth-class-taut:3:a")
2633 also AOT_have ‹… ≡ ¬¬(◇φ ∨ ◇ψ)›
2634 using "≡E"(6) "oth-class-taut:3:b" "oth-class-taut:5:b" by blast
2635 also AOT_have ‹… ≡ ◇φ ∨ ◇ψ›
2636 by (simp add: "≡I" "useful-tautologies:1" "useful-tautologies:2")
2637 finally show ?thesis .
2638qed
2639
2640AOT_theorem "KBasic2:3": ‹◇(φ & ψ) → (◇φ & ◇ψ)›
2641 by (metis "RM◇" "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "deduction-theorem" "modus-tollens:1" "reductio-aa:1")
2642
2643AOT_theorem "KBasic2:4": ‹◇(φ → ψ) ≡ (□φ → ◇ψ)›
2644proof -
2645 AOT_have ‹◇(φ → ψ) ≡ ◇(¬φ ∨ ψ)›
2646 by (AOT_subst ‹φ → ψ› ‹¬φ ∨ ψ›)
2647 (auto simp: "oth-class-taut:1:c" "oth-class-taut:3:a")
2648 also AOT_have ‹... ≡ ◇¬φ ∨ ◇ψ›
2649 by (simp add: "KBasic2:2")
2650 also AOT_have ‹... ≡ ¬□φ ∨ ◇ψ›
2651 by (AOT_subst ‹¬□φ› ‹◇¬φ›)
2652 (auto simp: "KBasic:11" "oth-class-taut:3:a")
2653 also AOT_have ‹... ≡ □φ → ◇ψ›
2654 using "≡E"(6) "oth-class-taut:1:c" "oth-class-taut:3:a" by blast
2655 finally show ?thesis .
2656qed
2657
2658AOT_theorem "KBasic2:5": ‹◇◇φ ≡ ¬□□¬φ›
2659 using "conventions:5"[THEN "≡Df"]
2660 by (AOT_subst ‹◇φ› ‹¬□¬φ›; AOT_subst ‹◇¬□¬φ› ‹¬□¬¬□¬φ›; AOT_subst (reverse) ‹¬¬□¬φ› ‹□¬φ›)
2661 (auto simp: "oth-class-taut:3:b" "oth-class-taut:3:a")
2662
2663
2664AOT_theorem "KBasic2:6": ‹□(φ ∨ ψ) → (□φ ∨ ◇ψ)›
2665proof(rule "→I"; rule "raa-cor:1")
2666 AOT_assume ‹□(φ ∨ ψ)›
2667 AOT_hence ‹□(¬φ → ψ)›
2668 using "conventions:2"[THEN "≡Df"]
2669 by (AOT_subst (reverse) ‹¬φ → ψ› ‹φ ∨ ψ›) simp
2670 AOT_hence 1: ‹◇¬φ → ◇ψ› using "KBasic:13" "vdash-properties:10" by blast
2671 AOT_assume ‹¬(□φ ∨ ◇ψ)›
2672 AOT_hence ‹¬□φ› and ‹¬◇ψ› using "&E" "≡E"(1) "oth-class-taut:5:d" by blast+
2673 AOT_thus ‹◇ψ & ¬◇ψ› using "&I"(1) 1[THEN "→E"] "KBasic:11" "≡E"(4) "raa-cor:3" by blast
2674qed
2675
2676AOT_theorem "KBasic2:7": ‹(□(φ ∨ ψ) & ◇¬φ) → ◇ψ›
2677proof(rule "→I"; frule "&E"(1); drule "&E"(2))
2678 AOT_assume ‹□(φ ∨ ψ)›
2679 AOT_hence 1: ‹□φ ∨ ◇ψ›
2680 using "KBasic2:6" "∨I"(2) "∨E"(1) by blast
2681 AOT_assume ‹◇¬φ›
2682 AOT_hence ‹¬□φ› using "KBasic:11" "≡E"(2) by blast
2683 AOT_thus ‹◇ψ› using 1 "∨E"(2) by blast
2684qed
2685
2686AOT_theorem "T-S5-fund:1": ‹φ → ◇φ›
2687 by (meson "≡⇩d⇩fI" "conventions:5" "contraposition:2" "Hypothetical Syllogism" "deduction-theorem" "qml:2"[axiom_inst])
2688lemmas "T◇" = "T-S5-fund:1"
2689
2690AOT_theorem "T-S5-fund:2": ‹◇□φ → □φ›
2691proof(rule "→I")
2692 AOT_assume ‹◇□φ›
2693 AOT_hence ‹¬□◇¬φ›
2694 using "KBasic:14" "≡E"(4) "raa-cor:3" by blast
2695 moreover AOT_have ‹◇¬φ → □◇¬φ›
2696 by (fact "qml:3"[axiom_inst])
2697 ultimately AOT_have ‹¬◇¬φ›
2698 using "modus-tollens:1" by blast
2699 AOT_thus ‹□φ› using "KBasic:12" "≡E"(2) by blast
2700qed
2701lemmas "5◇" = "T-S5-fund:2"
2702
2703
2704AOT_theorem "Act-Sub:1": ‹❙𝒜φ ≡ ¬❙𝒜¬φ›
2705 by (AOT_subst ‹❙𝒜¬φ› ‹¬❙𝒜φ›)
2706 (auto simp: "logic-actual-nec:1"[axiom_inst] "oth-class-taut:3:b")
2707
2708AOT_theorem "Act-Sub:2": ‹◇φ ≡ ❙𝒜◇φ›
2709 using "conventions:5"[THEN "≡Df"]
2710 by (AOT_subst ‹◇φ› ‹¬□¬φ›)
2711 (metis "deduction-theorem" "≡I" "≡E"(1) "≡E"(2) "≡E"(3)
2712 "logic-actual-nec:1"[axiom_inst] "qml-act:2"[axiom_inst])
2713
2714AOT_theorem "Act-Sub:3": ‹❙𝒜φ → ◇φ›
2715 using "conventions:5"[THEN "≡Df"]
2716 by (AOT_subst ‹◇φ› ‹¬□¬φ›)
2717 (metis "Act-Sub:1" "deduction-theorem" "≡E"(4) "nec-imp-act" "reductio-aa:2" "→E")
2718
2719
2720AOT_theorem "Act-Sub:4": ‹❙𝒜φ ≡ ◇❙𝒜φ›
2721proof (rule "≡I"; rule "→I")
2722 AOT_assume ‹❙𝒜φ›
2723 AOT_thus ‹◇❙𝒜φ› using "T◇" "vdash-properties:10" by blast
2724next
2725 AOT_assume ‹◇❙𝒜φ›
2726 AOT_hence ‹¬□¬❙𝒜φ›
2727 using "≡⇩d⇩fE" "conventions:5" by blast
2728 AOT_hence ‹¬□❙𝒜¬φ›
2729 by (AOT_subst ‹❙𝒜¬φ› ‹¬❙𝒜φ›)
2730 (simp add: "logic-actual-nec:1"[axiom_inst])
2731 AOT_thus ‹❙𝒜φ›
2732 using "Act-Basic:1" "Act-Basic:6" "∨E"(3) "≡E"(4) "reductio-aa:1" by blast
2733qed
2734
2735AOT_theorem "Act-Sub:5": ‹◇❙𝒜φ → ❙𝒜◇φ›
2736 by (metis "Act-Sub:2" "Act-Sub:3" "Act-Sub:4" "deduction-theorem" "≡E"(1) "≡E"(2) "vdash-properties:6")
2737
2738AOT_theorem "S5Basic:1": ‹◇φ ≡ □◇φ›
2739 by (simp add: "≡I" "qml:2" "qml:3" "vdash-properties:1[2]")
2740
2741AOT_theorem "S5Basic:2": ‹□φ ≡ ◇□φ›
2742 by (simp add: "T◇" "5◇" "≡I")
2743
2744AOT_theorem "S5Basic:3": ‹φ → □◇φ›
2745 using "T◇" "Hypothetical Syllogism" "qml:3" "vdash-properties:1[2]" by blast
2746lemmas "B" = "S5Basic:3"
2747
2748AOT_theorem "S5Basic:4": ‹◇□φ → φ›
2749 using "5◇" "Hypothetical Syllogism" "qml:2" "vdash-properties:1[2]" by blast
2750lemmas "B◇" = "S5Basic:4"
2751
2752AOT_theorem "S5Basic:5": ‹□φ → □□φ›
2753 using "RM:1" "B" "5◇" "Hypothetical Syllogism" by blast
2754lemmas "4" = "S5Basic:5"
2755
2756AOT_theorem "S5Basic:6": ‹□φ ≡ □□φ›
2757 by (simp add: "4" "≡I" "qml:2"[axiom_inst])
2758
2759AOT_theorem "S5Basic:7": ‹◇◇φ → ◇φ›
2760 using "conventions:5"[THEN "≡Df"] "oth-class-taut:3:b"
2761 by (AOT_subst ‹◇◇φ› ‹¬□¬◇φ›;
2762 AOT_subst ‹◇φ› ‹¬□¬φ›;
2763 AOT_subst (reverse) ‹¬¬□¬φ› ‹□¬φ›;
2764 AOT_subst (reverse) ‹□□¬φ› ‹□¬φ›)
2765 (auto simp: "S5Basic:6" "if-p-then-p")
2766
2767lemmas "4◇" = "S5Basic:7"
2768
2769AOT_theorem "S5Basic:8": ‹◇◇φ ≡ ◇φ›
2770 by (simp add: "4◇" "T◇" "≡I")
2771
2772AOT_theorem "S5Basic:9": ‹□(φ ∨ □ψ) ≡ (□φ ∨ □ψ)›
2773 apply (rule "≡I"; rule "→I")
2774 using "KBasic2:6" "5◇" "∨I"(3) "if-p-then-p" "vdash-properties:10" apply blast
2775 by (meson "KBasic:15" "4" "∨I"(3) "∨E"(1) "Disjunction Addition"(1) "con-dis-taut:7"
2776 "intro-elim:1" "Commutativity of ∨")
2777
2778AOT_theorem "S5Basic:10": ‹□(φ ∨ ◇ψ) ≡ (□φ ∨ ◇ψ)›
2779
2780proof(rule "≡I"; rule "→I")
2781 AOT_assume ‹□(φ ∨ ◇ψ)›
2782 AOT_hence ‹□φ ∨ ◇◇ψ›
2783 by (meson "KBasic2:6" "∨I"(2) "∨E"(1))
2784 AOT_thus ‹□φ ∨ ◇ψ›
2785 by (meson "B◇" "4" "4◇" "T◇" "∨I"(3))
2786next
2787 AOT_assume ‹□φ ∨ ◇ψ›
2788 AOT_hence ‹□φ ∨ □◇ψ›
2789 by (meson "S5Basic:1" "B◇" "S5Basic:6" "T◇" "5◇" "∨I"(3) "intro-elim:1")
2790 AOT_thus ‹□(φ ∨ ◇ψ)›
2791 by (meson "KBasic:15" "∨I"(3) "∨E"(1) "Disjunction Addition"(1) "Disjunction Addition"(2))
2792qed
2793
2794AOT_theorem "S5Basic:11": ‹◇(φ & ◇ψ) ≡ (◇φ & ◇ψ)›
2795proof -
2796 AOT_have ‹◇(φ & ◇ψ) ≡ ◇¬(¬φ ∨ ¬◇ψ)›
2797 by (AOT_subst ‹φ & ◇ψ› ‹¬(¬φ ∨ ¬◇ψ)›)
2798 (auto simp: "oth-class-taut:5:a" "oth-class-taut:3:a")
2799 also AOT_have ‹… ≡ ◇¬(¬φ ∨ □¬ψ)›
2800 by (AOT_subst ‹□¬ψ› ‹¬◇ψ›)
2801 (auto simp: "KBasic2:1" "oth-class-taut:3:a")
2802 also AOT_have ‹… ≡ ¬□(¬φ ∨ □¬ψ)›
2803 using "KBasic:11" "≡E"(6) "oth-class-taut:3:a" by blast
2804 also AOT_have ‹… ≡ ¬(□¬φ ∨ □¬ψ)›
2805 using "S5Basic:9" "≡E"(1) "oth-class-taut:4:b" by blast
2806 also AOT_have ‹… ≡ ¬(¬◇φ ∨ ¬◇ψ)›
2807 using "KBasic2:1"
2808 by (AOT_subst ‹□¬φ› ‹¬◇φ›; AOT_subst ‹□¬ψ› ‹¬◇ψ›)
2809 (auto simp: "oth-class-taut:3:a")
2810 also AOT_have ‹… ≡ ◇φ & ◇ψ›
2811 using "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:a" by blast
2812 finally show ?thesis .
2813qed
2814
2815AOT_theorem "S5Basic:12": ‹◇(φ & □ψ) ≡ (◇φ & □ψ)›
2816proof (rule "≡I"; rule "→I")
2817 AOT_assume ‹◇(φ & □ψ)›
2818 AOT_hence ‹◇φ & ◇□ψ›
2819 using "KBasic2:3" "vdash-properties:6" by blast
2820 AOT_thus ‹◇φ & □ψ›
2821 using "5◇" "&I" "&E"(1) "&E"(2) "vdash-properties:6" by blast
2822next
2823 AOT_assume ‹◇φ & □ψ›
2824 moreover AOT_have ‹(□□ψ & ◇φ) → ◇(φ & □ψ)›
2825 by (AOT_subst ‹φ & □ψ› ‹□ψ & φ›)
2826 (auto simp: "Commutativity of &" "KBasic:16")
2827 ultimately AOT_show ‹◇(φ & □ψ)›
2828 by (metis "4" "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "vdash-properties:6")
2829qed
2830
2831
2832AOT_theorem "S5Basic:13": ‹□(φ → □ψ) ≡ □(◇φ → ψ)›
2833proof (rule "≡I")
2834 AOT_modally_strict {
2835 AOT_have ‹□(φ → □ψ) → (◇φ → ψ)›
2836 by (meson "KBasic:13" "B◇" "Hypothetical Syllogism" "deduction-theorem")
2837 }
2838 AOT_hence ‹□□(φ → □ψ) → □(◇φ → ψ)›
2839 by (rule RM)
2840 AOT_thus ‹□(φ → □ψ) → □(◇φ → ψ)›
2841 using "4" "Hypothetical Syllogism" by blast
2842next
2843 AOT_modally_strict {
2844 AOT_have ‹□(◇φ → ψ) → (φ → □ψ)›
2845 by (meson "B" "Hypothetical Syllogism" "deduction-theorem" "qml:1" "vdash-properties:1[2]")
2846 }
2847 AOT_hence ‹□□(◇φ → ψ) → □(φ → □ψ)›
2848 by (rule RM)
2849 AOT_thus ‹□(◇φ → ψ) → □(φ → □ψ)›
2850 using "4" "Hypothetical Syllogism" by blast
2851qed
2852
2853AOT_theorem "derived-S5-rules:1":
2854 assumes ‹Γ ❙⊢⇩□ ◇φ → ψ› shows ‹□Γ ❙⊢⇩□ φ → □ψ›
2855proof -
2856 AOT_have ‹□Γ ❙⊢⇩□ □◇φ → □ψ›
2857 using assms by (rule "RM:1[prem]")
2858 AOT_thus ‹□Γ ❙⊢⇩□ φ → □ψ›
2859 using "B" "Hypothetical Syllogism" by blast
2860qed
2861
2862AOT_theorem "derived-S5-rules:2":
2863 assumes ‹Γ ❙⊢⇩□ φ → □ψ› shows ‹□Γ ❙⊢⇩□ ◇φ → ψ›
2864proof -
2865 AOT_have ‹□Γ ❙⊢⇩□ ◇φ → ◇□ψ›
2866 using assms by (rule "RM:2[prem]")
2867 AOT_thus ‹□Γ ❙⊢⇩□ ◇φ → ψ›
2868 using "B◇" "Hypothetical Syllogism" by blast
2869qed
2870
2871AOT_theorem "BFs:1": ‹∀α □φ{α} → □∀α φ{α}›
2872proof -
2873 AOT_modally_strict {
2874 AOT_modally_strict {
2875 AOT_have ‹∀α □φ{α} → □φ{α}› for α by (fact AOT)
2876 }
2877 AOT_hence ‹◇∀α □φ{α} → ◇□φ{α}› for α by (rule "RM◇")
2878 AOT_hence ‹◇∀α □φ{α} → ∀α φ{α}›
2879 using "B◇" "∀I" "→E" "→I" by metis
2880 }
2881 thus ?thesis using "derived-S5-rules:1" by blast
2882qed
2883lemmas "BF" = "BFs:1"
2884
2885AOT_theorem "BFs:2": ‹□∀α φ{α} → ∀α □φ{α}›
2886proof -
2887 AOT_have ‹□∀α φ{α} → □φ{α}› for α using RM "cqt-orig:3" by metis
2888 thus ?thesis using "cqt-orig:2"[THEN "→E"] "∀I" by metis
2889qed
2890lemmas "CBF" = "BFs:2"
2891
2892AOT_theorem "BFs:3": ‹◇∃α φ{α} → ∃α ◇φ{α}›
2893proof(rule "→I")
2894 AOT_modally_strict {
2895 AOT_have ‹□∀α ¬φ{α} ≡ ∀α □¬φ{α}›
2896 using BF CBF "≡I" by blast
2897 } note θ = this
2898
2899 AOT_assume ‹◇∃α φ{α}›
2900 AOT_hence ‹¬□¬(∃α φ{α})›
2901 using "≡⇩d⇩fE" "conventions:5" by blast
2902 AOT_hence ‹¬□∀α ¬φ{α}›
2903 apply (AOT_subst ‹∀α ¬φ{α}› ‹¬(∃α φ{α})›)
2904 using "≡⇩d⇩fI" "conventions:3" "conventions:4" "&I" "contraposition:2" "cqt-further:4"
2905 "df-rules-formulas[1]" "vdash-properties:1[2]" by blast
2906 AOT_hence ‹¬∀α □¬φ{α}›
2907 apply (AOT_subst (reverse) ‹∀α □¬φ{α}› ‹□∀α ¬φ{α}›)
2908 using θ by blast
2909 AOT_hence ‹¬∀α ¬¬□¬φ{α}›
2910 by (AOT_subst (reverse) ‹¬¬□¬φ{α}› ‹□¬φ{α}› bound: α)
2911 (simp add: "oth-class-taut:3:b")
2912 AOT_hence ‹∃α ¬□¬φ{α}›
2913 by (rule "conventions:4"[THEN "≡⇩d⇩fI"])
2914 AOT_thus ‹∃α ◇φ{α}›
2915 using "conventions:5"[THEN "≡Df"]
2916 by (AOT_subst ‹◇φ{α}› ‹¬□¬φ{α}› bound: α)
2917qed
2918lemmas "BF◇" = "BFs:3"
2919
2920AOT_theorem "BFs:4": ‹∃α ◇φ{α} → ◇∃α φ{α}›
2921proof(rule "→I")
2922 AOT_assume ‹∃α ◇φ{α}›
2923 AOT_hence ‹¬∀α ¬◇φ{α}›
2924 using "conventions:4"[THEN "≡⇩d⇩fE"] by blast
2925 AOT_hence ‹¬∀α □¬φ{α}›
2926 using "KBasic2:1"
2927 by (AOT_subst ‹□¬φ{α}› ‹¬◇φ{α}› bound: α)
2928 moreover AOT_have ‹∀α □¬φ{α} ≡ □∀α ¬φ{α}›
2929 using "≡I" "BF" "CBF" by metis
2930 ultimately AOT_have 1: ‹¬□∀α ¬φ{α}›
2931 using "≡E"(3) by blast
2932 AOT_show ‹◇∃α φ{α}›
2933 apply (rule "conventions:5"[THEN "≡⇩d⇩fI"])
2934 apply (AOT_subst ‹∃α φ{α}› ‹¬∀α ¬φ{α}›)
2935 apply (simp add: "conventions:4" "≡Df")
2936 apply (AOT_subst ‹¬¬∀α ¬φ{α}› ‹∀α ¬φ{α}›)
2937 by (auto simp: 1 "≡I" "useful-tautologies:1" "useful-tautologies:2")
2938qed
2939lemmas "CBF◇" = "BFs:4"
2940
2941AOT_theorem "sign-S5-thm:1": ‹∃α □φ{α} → □∃α φ{α}›
2942proof(rule "→I")
2943 AOT_assume ‹∃α □φ{α}›
2944 then AOT_obtain α where ‹□φ{α}› using "∃E" by metis
2945 moreover AOT_have ‹□α↓›
2946 by (simp add: "ex:1:a" "rule-ui:2[const_var]" RN)
2947 moreover AOT_have ‹□φ{τ}, □τ↓ ❙⊢⇩□ □∃α φ{α}› for τ
2948 proof -
2949 AOT_have ‹φ{τ}, τ↓ ❙⊢⇩□ ∃α φ{α}› using "existential:1" by blast
2950 AOT_thus ‹□φ{τ}, □τ↓ ❙⊢⇩□ □∃α φ{α}›
2951 using "RN[prem]"[where Γ="{φ τ, «τ↓»}", simplified] by blast
2952 qed
2953 ultimately AOT_show ‹□∃α φ{α}› by blast
2954qed
2955lemmas Buridan = "sign-S5-thm:1"
2956
2957AOT_theorem "sign-S5-thm:2": ‹◇∀α φ{α} → ∀α ◇φ{α}›
2958proof -
2959 AOT_have ‹∀α (◇∀α φ{α} → ◇φ{α})›
2960 by (simp add: "RM◇" "cqt-orig:3" "∀I")
2961 AOT_thus ‹◇∀α φ{α} → ∀α ◇φ{α}›
2962 using "∀E"(4) "∀I" "→E" "→I" by metis
2963qed
2964lemmas "Buridan◇" = "sign-S5-thm:2"
2965
2966AOT_theorem "sign-S5-thm:3": ‹◇∃α (φ{α} & ψ{α}) → ◇(∃α φ{α} & ∃α ψ{α})›
2967 apply (rule "RM:2")
2968 by (metis (no_types, lifting) "instantiation" "&I" "&E"(1)
2969 "&E"(2) "deduction-theorem" "existential:2[const_var]")
2970
2971AOT_theorem "sign-S5-thm:4": ‹◇∃α (φ{α} & ψ{α}) → ◇∃α φ{α}›
2972 apply (rule "RM:2")
2973 by (meson "instantiation" "&E"(1) "deduction-theorem" "existential:2[const_var]")
2974
2975AOT_theorem "sign-S5-thm:5": ‹(□∀α (φ{α} → ψ{α}) & □∀α (ψ{α} → χ{α})) → □∀α (φ{α} → χ{α})›
2976proof -
2977 {
2978 fix φ' ψ' χ'
2979 AOT_assume ‹❙⊢⇩□ φ' & ψ' → χ'›
2980 AOT_hence ‹□φ' & □ψ' → □χ'›
2981 using "RN[prem]"[where Γ="{φ', ψ'}"] apply simp
2982 using "&E" "&I" "→E" "→I" by metis
2983 } note R = this
2984 show ?thesis by (rule R; fact AOT)
2985qed
2986
2987AOT_theorem "sign-S5-thm:6": ‹(□∀α (φ{α} ≡ ψ{α}) & □∀α(ψ{α} ≡ χ{α})) → □∀α(φ{α} ≡ χ{α})›
2988proof -
2989 {
2990 fix φ' ψ' χ'
2991 AOT_assume ‹❙⊢⇩□ φ' & ψ' → χ'›
2992 AOT_hence ‹□φ' & □ψ' → □χ'›
2993 using "RN[prem]"[where Γ="{φ', ψ'}"] apply simp
2994 using "&E" "&I" "→E" "→I" by metis
2995 } note R = this
2996 show ?thesis by (rule R; fact AOT)
2997qed
2998
2999AOT_theorem "exist-nec2:1": ‹◇τ↓ → τ↓›
3000 using "B◇" "RM◇" "Hypothetical Syllogism" "exist-nec" by blast
3001
3002AOT_theorem "exists-nec2:2": ‹◇τ↓ ≡ □τ↓›
3003 by (meson "Act-Sub:3" "Hypothetical Syllogism" "exist-nec" "exist-nec2:1" "≡I" "nec-imp-act")
3004
3005AOT_theorem "exists-nec2:3": ‹¬τ↓ → □¬τ↓›
3006 using "KBasic2:1" "deduction-theorem" "exist-nec2:1" "≡E"(2) "modus-tollens:1" by blast
3007
3008AOT_theorem "exists-nec2:4": ‹◇¬τ↓ ≡ □¬τ↓›
3009 by (metis "Act-Sub:3" "KBasic:12" "deduction-theorem" "exist-nec" "exists-nec2:3" "≡I" "≡E"(4) "nec-imp-act" "reductio-aa:1")
3010
3011AOT_theorem "id-nec2:1": ‹◇α = β → α = β›
3012 using "B◇" "RM◇" "Hypothetical Syllogism" "id-nec:1" by blast
3013
3014AOT_theorem "id-nec2:2": ‹α ≠ β → □α ≠ β›
3015 apply (AOT_subst_using subst: "=-infix"[THEN "≡Df"])
3016 using "KBasic2:1" "deduction-theorem" "id-nec2:1" "≡E"(2) "modus-tollens:1" by blast
3017
3018AOT_theorem "id-nec2:3": ‹◇α ≠ β → α ≠ β›
3019 apply (AOT_subst_using subst: "=-infix"[THEN "≡Df"])
3020 by (metis "KBasic:11" "deduction-theorem" "id-nec:2" "≡E"(3) "reductio-aa:2" "vdash-properties:6")
3021
3022AOT_theorem "id-nec2:4": ‹◇α = β → □α = β›
3023 using "Hypothetical Syllogism" "id-nec2:1" "id-nec:1" by blast
3024
3025AOT_theorem "id-nec2:5": ‹◇α ≠ β → □α ≠ β›
3026 using "id-nec2:3" "id-nec2:2" "→I" "→E" by metis
3027
3028AOT_theorem "sc-eq-box-box:1": ‹□(φ → □φ) ≡ (◇φ → □φ)›
3029 apply (rule "≡I"; rule "→I")
3030 using "KBasic:13" "5◇" "Hypothetical Syllogism" "vdash-properties:10" apply blast
3031 by (metis "KBasic2:1" "KBasic:1" "KBasic:2" "S5Basic:13" "≡E"(2) "raa-cor:5" "vdash-properties:6")
3032
3033AOT_theorem "sc-eq-box-box:2": ‹(□(φ → □φ) ∨ (◇φ → □φ)) → (◇φ ≡ □φ)›
3034 by (metis "Act-Sub:3" "KBasic:13" "5◇" "∨E"(2) "deduction-theorem" "≡I" "nec-imp-act" "raa-cor:2" "vdash-properties:10")
3035
3036AOT_theorem "sc-eq-box-box:3": ‹□(φ → □φ) → (¬□φ ≡ □¬φ)›
3037proof (rule "→I"; rule "≡I"; rule "→I")
3038 AOT_assume ‹□(φ → □φ)›
3039 AOT_hence ‹◇φ → □φ› using "sc-eq-box-box:1" "≡E" by blast
3040 moreover AOT_assume ‹¬□φ›
3041 ultimately AOT_have ‹¬◇φ›
3042 using "modus-tollens:1" by blast
3043 AOT_thus ‹□¬φ›
3044 using "KBasic2:1" "≡E"(2) by blast
3045next
3046 AOT_assume ‹□(φ → □φ)›
3047 moreover AOT_assume ‹□¬φ›
3048 ultimately AOT_show ‹¬□φ›
3049 using "modus-tollens:1" "qml:2" "vdash-properties:10" "vdash-properties:1[2]" by blast
3050qed
3051
3052AOT_theorem "sc-eq-box-box:4": ‹(□(φ → □φ) & □(ψ → □ψ)) → ((□φ ≡ □ψ) → □(φ ≡ ψ))›
3053proof(rule "→I"; rule "→I")
3054 AOT_assume θ: ‹□(φ → □φ) & □(ψ → □ψ)›
3055 AOT_assume ξ: ‹□φ ≡ □ψ›
3056 AOT_hence ‹(□φ & □ψ) ∨ (¬□φ & ¬□ψ)›
3057 using "≡E"(4) "oth-class-taut:4:g" "raa-cor:3" by blast
3058 moreover {
3059 AOT_assume ‹□φ & □ψ›
3060 AOT_hence ‹□(φ ≡ ψ)›
3061 using "KBasic:3" "KBasic:8" "≡E"(2) "vdash-properties:10" by blast
3062 }
3063 moreover {
3064 AOT_assume ‹¬□φ & ¬□ψ›
3065 moreover AOT_have ‹¬□φ ≡ □¬φ› and ‹¬□ψ ≡ □¬ψ›
3066 using θ "Conjunction Simplification"(1) "Conjunction Simplification"(2) "sc-eq-box-box:3" "vdash-properties:10" by metis+
3067 ultimately AOT_have ‹□¬φ & □¬ψ›
3068 by (metis "&I" "Conjunction Simplification"(1) "Conjunction Simplification"(2) "≡E"(4) "modus-tollens:1" "raa-cor:3")
3069 AOT_hence ‹□(φ ≡ ψ)›
3070 using "KBasic:3" "KBasic:9" "≡E"(2) "vdash-properties:10" by blast
3071 }
3072 ultimately AOT_show ‹□(φ ≡ ψ)›
3073 using "∨E"(2) "reductio-aa:1" by blast
3074qed
3075
3076AOT_theorem "sc-eq-box-box:5": ‹(□(φ → □φ) & □(ψ → □ψ)) → □((φ ≡ ψ) → □(φ ≡ ψ))›
3077proof (rule "→I")
3078 AOT_assume ‹(□(φ → □φ) & □(ψ → □ψ))›
3079 AOT_hence ‹□(□(φ → □φ) & □(ψ → □ψ))›
3080 using 4[THEN "→E"] "&E" "&I" "KBasic:3" "≡E"(2) by metis
3081 moreover AOT_have ‹□(□(φ → □φ) & □(ψ → □ψ)) → □((φ ≡ ψ) → □(φ ≡ ψ))›
3082 proof (rule RM; rule "→I"; rule "→I")
3083 AOT_modally_strict {
3084 AOT_assume A: ‹(□(φ → □φ) & □(ψ → □ψ))›
3085 AOT_hence ‹φ → □φ› and ‹ψ → □ψ›
3086 using "&E" "qml:2"[axiom_inst] "→E" by blast+
3087 moreover AOT_assume ‹φ ≡ ψ›
3088 ultimately AOT_have ‹□φ ≡ □ψ›
3089 using "→E" "qml:2"[axiom_inst] "≡E" "≡I" by meson
3090 moreover AOT_have ‹(□φ ≡ □ψ) → □(φ ≡ ψ)›
3091 using A "sc-eq-box-box:4" "→E" by blast
3092 ultimately AOT_show ‹□(φ ≡ ψ)› using "→E" by blast
3093 }
3094 qed
3095 ultimately AOT_show ‹□((φ ≡ ψ) → □(φ ≡ ψ))› using "→E" by blast
3096qed
3097
3098AOT_theorem "sc-eq-box-box:6": ‹□(φ → □φ) → ((φ → □ψ) → □(φ → ψ))›
3099proof (rule "→I"; rule "→I"; rule "raa-cor:1")
3100 AOT_assume ‹¬□(φ → ψ)›
3101 AOT_hence ‹◇¬(φ → ψ)› by (metis "KBasic:11" "≡E"(1))
3102 AOT_hence ‹◇(φ & ¬ψ)›
3103 by (AOT_subst ‹φ & ¬ψ› ‹¬(φ → ψ)›)
3104 (meson "Commutativity of ≡" "≡E"(1) "oth-class-taut:1:b")
3105 AOT_hence ‹◇φ› and 2: ‹◇¬ψ› using "KBasic2:3"[THEN "→E"] "&E" by blast+
3106 moreover AOT_assume ‹□(φ → □φ)›
3107 ultimately AOT_have ‹□φ› by (metis "≡E"(1) "sc-eq-box-box:1" "→E")
3108 AOT_hence φ using "qml:2"[axiom_inst, THEN "→E"] by blast
3109 moreover AOT_assume ‹φ → □ψ›
3110 ultimately AOT_have ‹□ψ› using "→E" by blast
3111 moreover AOT_have ‹¬□ψ› using 2 "KBasic:12" "¬¬I" "intro-elim:3:d" by blast
3112 ultimately AOT_show ‹□ψ & ¬□ψ› using "&I" by blast
3113qed
3114
3115AOT_theorem "sc-eq-box-box:7": ‹□(φ → □φ) → ((φ → ❙𝒜ψ) → ❙𝒜(φ → ψ))›
3116proof (rule "→I"; rule "→I"; rule "raa-cor:1")
3117 AOT_assume ‹¬❙𝒜(φ → ψ)›
3118 AOT_hence ‹❙𝒜¬(φ → ψ)› by (metis "Act-Basic:1" "∨E"(2))
3119 AOT_hence ‹❙𝒜(φ & ¬ψ)›
3120 by (AOT_subst ‹φ & ¬ψ› ‹¬(φ → ψ)›)
3121 (meson "Commutativity of ≡" "≡E"(1) "oth-class-taut:1:b")
3122 AOT_hence ‹❙𝒜φ› and 2: ‹❙𝒜¬ψ› using "Act-Basic:2"[THEN "≡E"(1)] "&E" by blast+
3123 AOT_hence ‹◇φ› by (metis "Act-Sub:3" "→E")
3124 moreover AOT_assume ‹□(φ → □φ)›
3125 ultimately AOT_have ‹□φ› by (metis "≡E"(1) "sc-eq-box-box:1" "→E")
3126 AOT_hence φ using "qml:2"[axiom_inst, THEN "→E"] by blast
3127 moreover AOT_assume ‹φ → ❙𝒜ψ›
3128 ultimately AOT_have ‹❙𝒜ψ› using "→E" by blast
3129 moreover AOT_have ‹¬❙𝒜ψ› using 2 by (meson "Act-Sub:1" "≡E"(4) "raa-cor:3")
3130 ultimately AOT_show ‹❙𝒜ψ & ¬❙𝒜ψ› using "&I" by blast
3131qed
3132
3133AOT_theorem "sc-eq-fur:1": ‹◇❙𝒜φ ≡ □❙𝒜φ›
3134 using "Act-Basic:6" "Act-Sub:4" "≡E"(6) by blast
3135
3136AOT_theorem "sc-eq-fur:2": ‹□(φ → □φ) → (❙𝒜φ ≡ φ)›
3137 by (metis "B◇" "Act-Sub:3" "KBasic:13" "T◇" "Hypothetical Syllogism" "deduction-theorem" "≡I" "nec-imp-act")
3138
3139AOT_theorem "sc-eq-fur:3": ‹□∀x (φ{x} → □φ{x}) → (∃!x φ{x} → ❙ιx φ{x}↓)›
3140proof (rule "→I"; rule "→I")
3141 AOT_assume ‹□∀x (φ{x} → □φ{x})›
3142 AOT_hence A: ‹∀x □(φ{x} → □φ{x})› using CBF "→E" by blast
3143 AOT_assume ‹∃!x φ{x}›
3144 then AOT_obtain a where a_def: ‹φ{a} & ∀y (φ{y} → y = a)›
3145 using "∃E"[rotated 1, OF "uniqueness:1"[THEN "≡⇩d⇩fE"]] by blast
3146 moreover AOT_have ‹□φ{a}› using calculation A "∀E"(2) "qml:2"[axiom_inst] "→E" "&E"(1) by blast
3147 AOT_hence ‹❙𝒜φ{a}› using "nec-imp-act" "vdash-properties:6" by blast
3148 moreover AOT_have ‹∀y (❙𝒜φ{y} → y = a)›
3149 proof (rule "∀I"; rule "→I")
3150 fix b
3151 AOT_assume ‹❙𝒜φ{b}›
3152 AOT_hence ‹◇φ{b}›
3153 using "Act-Sub:3" "vdash-properties:6" by blast
3154 moreover {
3155 AOT_have ‹□(φ{b} → □φ{b})›
3156 using A "∀E"(2) by blast
3157 AOT_hence ‹◇φ{b} → □φ{b}›
3158 using "KBasic:13" "5◇" "Hypothetical Syllogism" "vdash-properties:6" by blast
3159 }
3160 ultimately AOT_have ‹□φ{b}› using "→E" by blast
3161 AOT_hence ‹φ{b}› using "qml:2"[axiom_inst] "→E" by blast
3162 AOT_thus ‹b = a›
3163 using a_def[THEN "&E"(2)] "∀E"(2) "→E" by blast
3164 qed
3165 ultimately AOT_have ‹❙𝒜φ{a} & ∀y (❙𝒜φ{y} → y = a)›
3166 using "&I" by blast
3167 AOT_hence ‹∃x (❙𝒜φ{x} & ∀y (❙𝒜φ{y} → y = x))› using "∃I" by fast
3168 AOT_hence ‹∃!x ❙𝒜φ{x}› using "uniqueness:1"[THEN "≡⇩d⇩fI"] by fast
3169 AOT_thus ‹❙ιx φ{x}↓›
3170 using "actual-desc:1"[THEN "≡E"(2)] by blast
3171qed
3172
3173AOT_theorem "sc-eq-fur:4": ‹□∀x (φ{x} → □φ{x}) → (x = ❙ιx φ{x} ≡ (φ{x} & ∀z (φ{z} → z = x)))›
3174proof (rule "→I")
3175 AOT_assume ‹□∀x (φ{x} → □φ{x})›
3176 AOT_hence ‹∀x □(φ{x} → □φ{x})› using CBF "→E" by blast
3177 AOT_hence A: ‹❙𝒜φ{α} ≡ φ{α}› for α using "sc-eq-fur:2" "∀E" "→E" by fast
3178 AOT_show ‹x = ❙ιx φ{x} ≡ (φ{x} & ∀z (φ{z} → z = x))›
3179 proof (rule "≡I"; rule "→I")
3180 AOT_assume ‹x = ❙ιx φ{x}›
3181 AOT_hence B: ‹❙𝒜φ{x} & ∀z (❙𝒜φ{z} → z = x)›
3182 using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
3183 AOT_show ‹φ{x} & ∀z (φ{z} → z = x)›
3184 proof (rule "&I"; (rule "∀I"; rule "→I")?)
3185 AOT_show ‹φ{x}› using A B[THEN "&E"(1)] "≡E"(1) by blast
3186 next
3187 AOT_show ‹z = x› if ‹φ{z}› for z
3188 using that B[THEN "&E"(2)] "∀E"(2) "→E" A[THEN "≡E"(2)] by blast
3189 qed
3190 next
3191 AOT_assume B: ‹φ{x} & ∀z (φ{z} → z = x)›
3192 AOT_have ‹❙𝒜φ{x} & ∀z (❙𝒜φ{z} → z = x)›
3193 proof(rule "&I"; (rule "∀I"; rule "→I")?)
3194 AOT_show ‹❙𝒜φ{x}› using B[THEN "&E"(1)] A[THEN "≡E"(2)] by blast
3195 next
3196 AOT_show ‹b = x› if ‹❙𝒜φ{b}› for b
3197 using that A[THEN "≡E"(1)] B[THEN "&E"(2), THEN "∀E"(2), THEN "→E"] by blast
3198 qed
3199 AOT_thus ‹x = ❙ιx φ{x}›
3200 using "nec-hintikka-scheme"[THEN "≡E"(2)] by blast
3201 qed
3202qed
3203
3204AOT_theorem "id-act:1": ‹α = β ≡ ❙𝒜α = β›
3205 by (meson "Act-Sub:3" "Hypothetical Syllogism" "id-nec2:1" "id-nec:2" "≡I" "nec-imp-act")
3206
3207AOT_theorem "id-act:2": ‹α ≠ β ≡ ❙𝒜α ≠ β›
3208proof (AOT_subst ‹α ≠ β› ‹¬(α = β)›)
3209 AOT_modally_strict {
3210 AOT_show ‹α ≠ β ≡ ¬(α = β)›
3211 by (simp add: "=-infix" "≡Df")
3212 }
3213next
3214 AOT_show ‹¬(α = β) ≡ ❙𝒜¬(α = β)›
3215 proof (safe intro!: "≡I" "→I")
3216 AOT_assume ‹¬α = β›
3217 AOT_hence ‹¬❙𝒜α = β› using "id-act:1" "≡E"(3) by blast
3218 AOT_thus ‹❙𝒜¬α = β›
3219 using "¬¬E" "Act-Sub:1" "≡E"(3) by blast
3220 next
3221 AOT_assume ‹❙𝒜¬α = β›
3222 AOT_hence ‹¬❙𝒜α = β›
3223 using "¬¬I" "Act-Sub:1" "≡E"(4) by blast
3224 AOT_thus ‹¬α = β›
3225 using "id-act:1" "≡E"(4) by blast
3226 qed
3227qed
3228
3229AOT_theorem "A-Exists:1": ‹❙𝒜∃!α φ{α} ≡ ∃!α ❙𝒜φ{α}›
3230proof -
3231 AOT_have ‹❙𝒜∃!α φ{α} ≡ ❙𝒜∃α∀β (φ{β} ≡ β = α)›
3232 by (AOT_subst_using subst: "uniqueness:2")
3233 (simp add: "oth-class-taut:3:a")
3234 also AOT_have ‹… ≡ ∃α ❙𝒜∀β (φ{β} ≡ β = α)›
3235 by (simp add: "Act-Basic:10")
3236 also AOT_have ‹… ≡ ∃α∀β ❙𝒜(φ{β} ≡ β = α)›
3237 by (AOT_subst ‹❙𝒜∀β (φ{β} ≡ β = α)› ‹∀β ❙𝒜(φ{β} ≡ β = α)› bound: α)
3238 (auto simp: "logic-actual-nec:3" "vdash-properties:1[2]" "oth-class-taut:3:a")
3239 also AOT_have ‹… ≡ ∃α∀β (❙𝒜φ{β} ≡ ❙𝒜β = α)›
3240 by (AOT_subst_old_rev "λ τ τ' . «❙𝒜(φ{τ'} ≡ τ' = τ)»" "λ τ τ'. «❙𝒜φ{τ'} ≡ ❙𝒜τ' = τ»")
3241 (auto simp: "Act-Basic:5" "cqt-further:7")
3242 also AOT_have ‹… ≡ ∃α∀β (❙𝒜φ{β} ≡ β = α)›
3243
3244 apply (AOT_subst_old "λ τ τ' :: 'a . «❙𝒜τ' = τ»" "λ τ τ'. «τ' = τ»")
3245 apply (meson "id-act:1" "≡E"(6) "oth-class-taut:3:a")
3246 by (simp add: "cqt-further:7")
3247 also AOT_have ‹... ≡ ∃!α ❙𝒜φ{α}›
3248 using "uniqueness:2" "Commutativity of ≡"[THEN "≡E"(1)] by fast
3249 finally show ?thesis .
3250qed
3251
3252AOT_theorem "A-Exists:2": ‹❙ιx φ{x}↓ ≡ ❙𝒜∃!x φ{x}›
3253 by (AOT_subst_using subst: "A-Exists:1")
3254 (simp add: "actual-desc:1")
3255
3256AOT_theorem "id-act-desc:1": ‹❙ιx (x = y)↓›
3257proof(rule "existence:1"[THEN "≡⇩d⇩fI"]; rule "∃I")
3258 AOT_show ‹[λx E!x → E!x]❙ιx (x = y)›
3259 proof (rule "russell-axiom[exe,1].nec-russell-axiom"[THEN "≡E"(2)]; rule "∃I"; (rule "&I")+)
3260 AOT_show ‹❙𝒜y = y› by (simp add: "RA[2]" "id-eq:1")
3261 next
3262 AOT_show ‹∀z (❙𝒜z = y → z = y)›
3263 apply (rule "∀I")
3264 using "id-act:1"[THEN "≡E"(2)] "→I" by blast
3265 next
3266 AOT_show ‹[λx E!x → E!x]y›
3267 proof (rule "lambda-predicates:2"[axiom_inst, THEN "→E", THEN "≡E"(2)])
3268 AOT_show ‹[λx E!x → E!x]↓›
3269 by "cqt:2[lambda]"
3270 next
3271 AOT_show ‹E!y → E!y›
3272 by (simp add: "if-p-then-p")
3273 qed
3274 qed
3275next
3276 AOT_show ‹[λx E!x → E!x]↓›
3277 by "cqt:2[lambda]"
3278qed
3279
3280AOT_theorem "id-act-desc:2": ‹y = ❙ιx (x = y)›
3281 by (rule descriptions[axiom_inst, THEN "≡E"(2)]; rule "∀I"; rule "id-act:1"[symmetric])
3282
3283AOT_theorem "pre-en-eq:1[1]": ‹x⇩1[F] → □x⇩1[F]›
3284 by (simp add: encoding "vdash-properties:1[2]")
3285
3286AOT_theorem "pre-en-eq:1[2]": ‹x⇩1x⇩2[F] → □x⇩1x⇩2[F]›
3287proof (rule "→I")
3288 AOT_assume ‹x⇩1x⇩2[F]›
3289 AOT_hence ‹x⇩1[λy [F]yx⇩2]› and ‹x⇩2[λy [F]x⇩1y]›
3290 using "nary-encoding[2]"[axiom_inst, THEN "≡E"(1)] "&E" by blast+
3291 moreover AOT_have ‹[λy [F]yx⇩2]↓› by "cqt:2[lambda]"
3292 moreover AOT_have ‹[λy [F]x⇩1y]↓› by "cqt:2[lambda]"
3293 ultimately AOT_have ‹□x⇩1[λy [F]yx⇩2]› and ‹□x⇩2[λy [F]x⇩1y]›
3294 using encoding[axiom_inst, unvarify F] "→E" "&I" by blast+
3295 note A = this
3296 AOT_hence ‹□(x⇩1[λy [F]yx⇩2] & x⇩2[λy [F]x⇩1y])›
3297 using "KBasic:3"[THEN "≡E"(2)] "&I" by blast
3298 AOT_thus ‹□x⇩1x⇩2[F]›
3299 by (rule "nary-encoding[2]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3300qed
3301
3302AOT_theorem "pre-en-eq:1[3]": ‹x⇩1x⇩2x⇩3[F] → □x⇩1x⇩2x⇩3[F]›
3303proof (rule "→I")
3304 AOT_assume ‹x⇩1x⇩2x⇩3[F]›
3305 AOT_hence ‹x⇩1[λy [F]yx⇩2x⇩3]› and ‹x⇩2[λy [F]x⇩1yx⇩3]› and ‹x⇩3[λy [F]x⇩1x⇩2y]›
3306 using "nary-encoding[3]"[axiom_inst, THEN "≡E"(1)] "&E" by blast+
3307 moreover AOT_have ‹[λy [F]yx⇩2x⇩3]↓› by "cqt:2[lambda]"
3308 moreover AOT_have ‹[λy [F]x⇩1yx⇩3]↓› by "cqt:2[lambda]"
3309 moreover AOT_have ‹[λy [F]x⇩1x⇩2y]↓› by "cqt:2[lambda]"
3310 ultimately AOT_have ‹□x⇩1[λy [F]yx⇩2x⇩3]› and ‹□x⇩2[λy [F]x⇩1yx⇩3]› and ‹□x⇩3[λy [F]x⇩1x⇩2y]›
3311 using encoding[axiom_inst, unvarify F] "→E" by blast+
3312 note A = this
3313 AOT_have B: ‹□(x⇩1[λy [F]yx⇩2x⇩3] & x⇩2[λy [F]x⇩1yx⇩3] & x⇩3[λy [F]x⇩1x⇩2y])›
3314 by (rule "KBasic:3"[THEN "≡E"(2)] "&I" A)+
3315 AOT_thus ‹□x⇩1x⇩2x⇩3[F]›
3316 by (rule "nary-encoding[3]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3317qed
3318
3319AOT_theorem "pre-en-eq:1[4]": ‹x⇩1x⇩2x⇩3x⇩4[F] → □x⇩1x⇩2x⇩3x⇩4[F]›
3320proof (rule "→I")
3321 AOT_assume ‹x⇩1x⇩2x⇩3x⇩4[F]›
3322 AOT_hence ‹x⇩1[λy [F]yx⇩2x⇩3x⇩4]› and ‹x⇩2[λy [F]x⇩1yx⇩3x⇩4]› and ‹x⇩3[λy [F]x⇩1x⇩2yx⇩4]› and ‹x⇩4[λy [F]x⇩1x⇩2x⇩3y]›
3323 using "nary-encoding[4]"[axiom_inst, THEN "≡E"(1)] "&E" by metis+
3324 moreover AOT_have ‹[λy [F]yx⇩2x⇩3x⇩4]↓› by "cqt:2[lambda]"
3325 moreover AOT_have ‹[λy [F]x⇩1yx⇩3x⇩4]↓› by "cqt:2[lambda]"
3326 moreover AOT_have ‹[λy [F]x⇩1x⇩2yx⇩4]↓› by "cqt:2[lambda]"
3327 moreover AOT_have ‹[λy [F]x⇩1x⇩2x⇩3y]↓› by "cqt:2[lambda]"
3328 ultimately AOT_have ‹□x⇩1[λy [F]yx⇩2x⇩3x⇩4]› and ‹□x⇩2[λy [F]x⇩1yx⇩3x⇩4]› and ‹□x⇩3[λy [F]x⇩1x⇩2yx⇩4]› and ‹□x⇩4[λy [F]x⇩1x⇩2x⇩3y]›
3329 using "→E" encoding[axiom_inst, unvarify F] by blast+
3330 note A = this
3331 AOT_have B: ‹□(x⇩1[λy [F]yx⇩2x⇩3x⇩4] & x⇩2[λy [F]x⇩1yx⇩3x⇩4] & x⇩3[λy [F]x⇩1x⇩2yx⇩4] & x⇩4[λy [F]x⇩1x⇩2x⇩3y])›
3332 by (rule "KBasic:3"[THEN "≡E"(2)] "&I" A)+
3333 AOT_thus ‹□x⇩1x⇩2x⇩3x⇩4[F]›
3334 by (rule "nary-encoding[4]"[axiom_inst, THEN RN, THEN "KBasic:6"[THEN "→E"], THEN "≡E"(2)])
3335qed
3336
3337AOT_theorem "pre-en-eq:2[1]": ‹¬x⇩1[F] → □¬x⇩1[F]›
3338proof (rule "→I"; rule "raa-cor:1")
3339 AOT_assume ‹¬□¬x⇩1[F]›
3340 AOT_hence ‹◇x⇩1[F]›
3341 by (rule "conventions:5"[THEN "≡⇩d⇩fI"])
3342 AOT_hence ‹x⇩1[F]›
3343 by(rule "S5Basic:13"[THEN "≡E"(1), OF "pre-en-eq:1[1]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3344 moreover AOT_assume ‹¬x⇩1[F]›
3345 ultimately AOT_show ‹x⇩1[F] & ¬x⇩1[F]› by (rule "&I")
3346qed
3347AOT_theorem "pre-en-eq:2[2]": ‹¬x⇩1x⇩2[F] → □¬x⇩1x⇩2[F]›
3348proof (rule "→I"; rule "raa-cor:1")
3349 AOT_assume ‹¬□¬x⇩1x⇩2[F]›
3350 AOT_hence ‹◇x⇩1x⇩2[F]›
3351 by (rule "conventions:5"[THEN "≡⇩d⇩fI"])
3352 AOT_hence ‹x⇩1x⇩2[F]›
3353 by(rule "S5Basic:13"[THEN "≡E"(1), OF "pre-en-eq:1[2]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3354 moreover AOT_assume ‹¬x⇩1x⇩2[F]›
3355 ultimately AOT_show ‹x⇩1x⇩2[F] & ¬x⇩1x⇩2[F]› by (rule "&I")
3356qed
3357
3358AOT_theorem "pre-en-eq:2[3]": ‹¬x⇩1x⇩2x⇩3[F] → □¬x⇩1x⇩2x⇩3[F]›
3359proof (rule "→I"; rule "raa-cor:1")
3360 AOT_assume ‹¬□¬x⇩1x⇩2x⇩3[F]›
3361 AOT_hence ‹◇x⇩1x⇩2x⇩3[F]›
3362 by (rule "conventions:5"[THEN "≡⇩d⇩fI"])
3363 AOT_hence ‹x⇩1x⇩2x⇩3[F]›
3364 by(rule "S5Basic:13"[THEN "≡E"(1), OF "pre-en-eq:1[3]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3365 moreover AOT_assume ‹¬x⇩1x⇩2x⇩3[F]›
3366 ultimately AOT_show ‹x⇩1x⇩2x⇩3[F] & ¬x⇩1x⇩2x⇩3[F]› by (rule "&I")
3367qed
3368
3369AOT_theorem "pre-en-eq:2[4]": ‹¬x⇩1x⇩2x⇩3x⇩4[F] → □¬x⇩1x⇩2x⇩3x⇩4[F]›
3370proof (rule "→I"; rule "raa-cor:1")
3371 AOT_assume ‹¬□¬x⇩1x⇩2x⇩3x⇩4[F]›
3372 AOT_hence ‹◇x⇩1x⇩2x⇩3x⇩4[F]›
3373 by (rule "conventions:5"[THEN "≡⇩d⇩fI"])
3374 AOT_hence ‹x⇩1x⇩2x⇩3x⇩4[F]›
3375 by(rule "S5Basic:13"[THEN "≡E"(1), OF "pre-en-eq:1[4]"[THEN RN], THEN "qml:2"[axiom_inst, THEN "→E"], THEN "→E"])
3376 moreover AOT_assume ‹¬x⇩1x⇩2x⇩3x⇩4[F]›
3377 ultimately AOT_show ‹x⇩1x⇩2x⇩3x⇩4[F] & ¬x⇩1x⇩2x⇩3x⇩4[F]› by (rule "&I")
3378qed
3379
3380AOT_theorem "en-eq:1[1]": ‹◇x⇩1[F] ≡ □x⇩1[F]›
3381 using "pre-en-eq:1[1]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by metis
3382AOT_theorem "en-eq:1[2]": ‹◇x⇩1x⇩2[F] ≡ □x⇩1x⇩2[F]›
3383 using "pre-en-eq:1[2]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by metis
3384AOT_theorem "en-eq:1[3]": ‹◇x⇩1x⇩2x⇩3[F] ≡ □x⇩1x⇩2x⇩3[F]›
3385 using "pre-en-eq:1[3]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by fast
3386AOT_theorem "en-eq:1[4]": ‹◇x⇩1x⇩2x⇩3x⇩4[F] ≡ □x⇩1x⇩2x⇩3x⇩4[F]›
3387 using "pre-en-eq:1[4]"[THEN RN] "sc-eq-box-box:2" "∨I" "→E" by fast
3388
3389AOT_theorem "en-eq:2[1]": ‹x⇩1[F] ≡ □x⇩1[F]›
3390 by (simp add: "≡I" "pre-en-eq:1[1]" "qml:2"[axiom_inst])
3391AOT_theorem "en-eq:2[2]": ‹x⇩1x⇩2[F] ≡ □x⇩1x⇩2[F]›
3392 by (simp add: "≡I" "pre-en-eq:1[2]" "qml:2"[axiom_inst])
3393AOT_theorem "en-eq:2[3]": ‹x⇩1x⇩2x⇩3[F] ≡ □x⇩1x⇩2x⇩3[F]›
3394 by (simp add: "≡I" "pre-en-eq:1[3]" "qml:2"[axiom_inst])
3395AOT_theorem "en-eq:2[4]": ‹x⇩1x⇩2x⇩3x⇩4[F] ≡ □x⇩1x⇩2x⇩3x⇩4[F]›
3396 by (simp add: "≡I" "pre-en-eq:1[4]" "qml:2"[axiom_inst])
3397
3398AOT_theorem "en-eq:3[1]": ‹◇x⇩1[F] ≡ x⇩1[F]›
3399 using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[1]"] "≡I" by blast
3400AOT_theorem "en-eq:3[2]": ‹◇x⇩1x⇩2[F] ≡ x⇩1x⇩2[F]›
3401 using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[2]"] "≡I" by blast
3402AOT_theorem "en-eq:3[3]": ‹◇x⇩1x⇩2x⇩3[F] ≡ x⇩1x⇩2x⇩3[F]›
3403 using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[3]"] "≡I" by blast
3404AOT_theorem "en-eq:3[4]": ‹◇x⇩1x⇩2x⇩3x⇩4[F] ≡ x⇩1x⇩2x⇩3x⇩4[F]›
3405 using "T◇" "derived-S5-rules:2"[where Γ="{}", OF "pre-en-eq:1[4]"] "≡I" by blast
3406
3407AOT_theorem "en-eq:4[1]": ‹(x⇩1[F] ≡ y⇩1[G]) ≡ (□x⇩1[F] ≡ □y⇩1[G])›
3408 apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3409 using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[1]" by blast+
3410AOT_theorem "en-eq:4[2]": ‹(x⇩1x⇩2[F] ≡ y⇩1y⇩2[G]) ≡ (□x⇩1x⇩2[F] ≡ □y⇩1y⇩2[G])›
3411 apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3412 using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[2]" by blast+
3413AOT_theorem "en-eq:4[3]": ‹(x⇩1x⇩2x⇩3[F] ≡ y⇩1y⇩2y⇩3[G]) ≡ (□x⇩1x⇩2x⇩3[F] ≡ □y⇩1y⇩2y⇩3[G])›
3414 apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3415 using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[3]" by blast+
3416AOT_theorem "en-eq:4[4]": ‹(x⇩1x⇩2x⇩3x⇩4[F] ≡ y⇩1y⇩2y⇩3y⇩4[G]) ≡ (□x⇩1x⇩2x⇩3x⇩4[F] ≡ □y⇩1y⇩2y⇩3y⇩4[G])›
3417 apply (rule "≡I"; rule "→I"; rule "≡I"; rule "→I")
3418 using "qml:2"[axiom_inst, THEN "→E"] "≡E"(1,2) "en-eq:2[4]" by blast+
3419
3420AOT_theorem "en-eq:5[1]": ‹□(x⇩1[F] ≡ y⇩1[G]) ≡ (□x⇩1[F] ≡ □y⇩1[G])›
3421 apply (rule "≡I"; rule "→I")
3422 using "en-eq:4[1]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3423 using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3424 "&I"[OF "pre-en-eq:1[1]"[THEN RN], OF "pre-en-eq:1[1]"[THEN RN]] by blast
3425AOT_theorem "en-eq:5[2]": ‹□(x⇩1x⇩2[F] ≡ y⇩1y⇩2[G]) ≡ (□x⇩1x⇩2[F] ≡ □y⇩1y⇩2[G])›
3426 apply (rule "≡I"; rule "→I")
3427 using "en-eq:4[2]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3428 using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3429 "&I"[OF "pre-en-eq:1[2]"[THEN RN], OF "pre-en-eq:1[2]"[THEN RN]] by blast
3430AOT_theorem "en-eq:5[3]": ‹□(x⇩1x⇩2x⇩3[F] ≡ y⇩1y⇩2y⇩3[G]) ≡ (□x⇩1x⇩2x⇩3[F] ≡ □y⇩1y⇩2y⇩3[G])›
3431 apply (rule "≡I"; rule "→I")
3432 using "en-eq:4[3]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3433 using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3434 "&I"[OF "pre-en-eq:1[3]"[THEN RN], OF "pre-en-eq:1[3]"[THEN RN]] by blast
3435AOT_theorem "en-eq:5[4]": ‹□(x⇩1x⇩2x⇩3x⇩4[F] ≡ y⇩1y⇩2y⇩3y⇩4[G]) ≡ (□x⇩1x⇩2x⇩3x⇩4[F] ≡ □y⇩1y⇩2y⇩3y⇩4[G])›
3436 apply (rule "≡I"; rule "→I")
3437 using "en-eq:4[4]"[THEN "≡E"(1)] "qml:2"[axiom_inst, THEN "→E"] apply blast
3438 using "sc-eq-box-box:4"[THEN "→E", THEN "→E"]
3439 "&I"[OF "pre-en-eq:1[4]"[THEN RN], OF "pre-en-eq:1[4]"[THEN RN]] by blast
3440
3441AOT_theorem "en-eq:6[1]": ‹(x⇩1[F] ≡ y⇩1[G]) ≡ □(x⇩1[F] ≡ y⇩1[G])›
3442 using "en-eq:5[1]"[symmetric] "en-eq:4[1]" "≡E"(5) by fast
3443AOT_theorem "en-eq:6[2]": ‹(x⇩1x⇩2[F] ≡ y⇩1y⇩2[G]) ≡ □(x⇩1x⇩2[F] ≡ y⇩1y⇩2[G])›
3444 using "en-eq:5[2]"[symmetric] "en-eq:4[2]" "≡E"(5) by fast
3445AOT_theorem "en-eq:6[3]": ‹(x⇩1x⇩2x⇩3[F] ≡ y⇩1y⇩2y⇩3[G]) ≡ □(x⇩1x⇩2x⇩3[F] ≡ y⇩1y⇩2y⇩3[G])›
3446 using "en-eq:5[3]"[symmetric] "en-eq:4[3]" "≡E"(5) by fast
3447AOT_theorem "en-eq:6[4]": ‹(x⇩1x⇩2x⇩3x⇩4[F] ≡ y⇩1y⇩2y⇩3y⇩4[G]) ≡ □(x⇩1x⇩2x⇩3x⇩4[F] ≡ y⇩1y⇩2y⇩3y⇩4[G])›
3448 using "en-eq:5[4]"[symmetric] "en-eq:4[4]" "≡E"(5) by fast
3449
3450AOT_theorem "en-eq:7[1]": ‹¬x⇩1[F] ≡ □¬x⇩1[F]›
3451 using "pre-en-eq:2[1]" "qml:2"[axiom_inst] "≡I" by blast
3452AOT_theorem "en-eq:7[2]": ‹¬x⇩1x⇩2[F] ≡ □¬x⇩1x⇩2[F]›
3453 using "pre-en-eq:2[2]" "qml:2"[axiom_inst] "≡I" by blast
3454AOT_theorem "en-eq:7[3]": ‹¬x⇩1x⇩2x⇩3[F] ≡ □¬x⇩1x⇩2x⇩3[F]›
3455 using "pre-en-eq:2[3]" "qml:2"[axiom_inst] "≡I" by blast
3456AOT_theorem "en-eq:7[4]": ‹¬x⇩1x⇩2x⇩3x⇩4[F] ≡ □¬x⇩1x⇩2x⇩3x⇩4[F]›
3457 using "pre-en-eq:2[4]" "qml:2"[axiom_inst] "≡I" by blast
3458
3459AOT_theorem "en-eq:8[1]": ‹◇¬x⇩1[F] ≡ ¬x⇩1[F]›
3460 using "en-eq:2[1]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3461AOT_theorem "en-eq:8[2]": ‹◇¬x⇩1x⇩2[F] ≡ ¬x⇩1x⇩2[F]›
3462 using "en-eq:2[2]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3463AOT_theorem "en-eq:8[3]": ‹◇¬x⇩1x⇩2x⇩3[F] ≡ ¬x⇩1x⇩2x⇩3[F]›
3464 using "en-eq:2[3]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3465AOT_theorem "en-eq:8[4]": ‹◇¬x⇩1x⇩2x⇩3x⇩4[F] ≡ ¬x⇩1x⇩2x⇩3x⇩4[F]›
3466 using "en-eq:2[4]"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "KBasic:11" "≡E"(5)[symmetric] by blast
3467
3468AOT_theorem "en-eq:9[1]": ‹◇¬x⇩1[F] ≡ □¬x⇩1[F]›
3469 using "en-eq:7[1]" "en-eq:8[1]" "≡E"(5) by blast
3470AOT_theorem "en-eq:9[2]": ‹◇¬x⇩1x⇩2[F] ≡ □¬x⇩1x⇩2[F]›
3471 using "en-eq:7[2]" "en-eq:8[2]" "≡E"(5) by blast
3472AOT_theorem "en-eq:9[3]": ‹◇¬x⇩1x⇩2x⇩3[F] ≡ □¬x⇩1x⇩2x⇩3[F]›
3473 using "en-eq:7[3]" "en-eq:8[3]" "≡E"(5) by blast
3474AOT_theorem "en-eq:9[4]": ‹◇¬x⇩1x⇩2x⇩3x⇩4[F] ≡ □¬x⇩1x⇩2x⇩3x⇩4[F]›
3475 using "en-eq:7[4]" "en-eq:8[4]" "≡E"(5) by blast
3476
3477AOT_theorem "en-eq:10[1]": ‹❙𝒜x⇩1[F] ≡ x⇩1[F]›
3478 by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[1]" "pre-en-eq:1[1]")
3479AOT_theorem "en-eq:10[2]": ‹❙𝒜x⇩1x⇩2[F] ≡ x⇩1x⇩2[F]›
3480 by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[2]" "pre-en-eq:1[2]")
3481AOT_theorem "en-eq:10[3]": ‹❙𝒜x⇩1x⇩2x⇩3[F] ≡ x⇩1x⇩2x⇩3[F]›
3482 by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[3]" "pre-en-eq:1[3]")
3483AOT_theorem "en-eq:10[4]": ‹❙𝒜x⇩1x⇩2x⇩3x⇩4[F] ≡ x⇩1x⇩2x⇩3x⇩4[F]›
3484 by (metis "Act-Sub:3" "deduction-theorem" "≡I" "≡E"(1) "nec-imp-act" "en-eq:3[4]" "pre-en-eq:1[4]")
3485
3486AOT_theorem "oa-facts:1": ‹O!x → □O!x›
3487proof(rule "→I")
3488 AOT_modally_strict {
3489 AOT_have ‹[λx ◇E!x]x ≡ ◇E!x›
3490 by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
3491 } note θ = this
3492 AOT_assume ‹O!x›
3493 AOT_hence ‹[λx ◇E!x]x›
3494 by (rule "=⇩d⇩fE"(2)[OF AOT_ordinary, rotated 1]) "cqt:2[lambda]"
3495 AOT_hence ‹◇E!x› using θ[THEN "≡E"(1)] by blast
3496 AOT_hence 0: ‹□◇E!x› using "qml:3"[axiom_inst, THEN "→E"] by blast
3497 AOT_have ‹□[λx ◇E!x]x›
3498 by (AOT_subst_using subst: θ) (simp add: 0)
3499 AOT_thus ‹□O!x›
3500 by (rule "=⇩d⇩fI"(2)[OF AOT_ordinary, rotated 1]) "cqt:2[lambda]"
3501qed
3502
3503AOT_theorem "oa-facts:2": ‹A!x → □A!x›
3504proof(rule "→I")
3505 AOT_modally_strict {
3506 AOT_have ‹[λx ¬◇E!x]x ≡ ¬◇E!x›
3507 by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2[lambda]"
3508 } note θ = this
3509 AOT_assume ‹A!x›
3510 AOT_hence ‹[λx ¬◇E!x]x›
3511 by (rule "=⇩d⇩fE"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
3512 AOT_hence ‹¬◇E!x› using θ[THEN "≡E"(1)] by blast
3513 AOT_hence ‹□¬E!x› using "KBasic2:1"[THEN "≡E"(2)] by blast
3514 AOT_hence 0: ‹□□¬E!x› using "4"[THEN "→E"] by blast
3515 AOT_have 1: ‹□¬◇E!x›
3516 apply (AOT_subst ‹¬◇E!x› ‹□¬E!x›)
3517 using "KBasic2:1"[symmetric] apply blast
3518 using 0 by blast
3519 AOT_have ‹□[λx ¬◇E!x]x›
3520 by (AOT_subst_using subst: θ) (simp add: 1)
3521 AOT_thus ‹□A!x›
3522 by (rule "=⇩d⇩fI"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
3523qed
3524
3525AOT_theorem "oa-facts:3": ‹◇O!x → O!x›
3526 using "oa-facts:1" "B◇" "RM◇" "Hypothetical Syllogism" by blast
3527AOT_theorem "oa-facts:4": ‹◇A!x → A!x›
3528 using "oa-facts:2" "B◇" "RM◇" "Hypothetical Syllogism" by blast
3529
3530AOT_theorem "oa-facts:5": ‹◇O!x ≡ □O!x›
3531 by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:1" "oa-facts:3")
3532
3533AOT_theorem "oa-facts:6": ‹◇A!x ≡ □A!x›
3534 by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:2" "oa-facts:4")
3535
3536AOT_theorem "oa-facts:7": ‹O!x ≡ ❙𝒜O!x›
3537 by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:1" "oa-facts:3")
3538
3539AOT_theorem "oa-facts:8": ‹A!x ≡ ❙𝒜A!x›
3540 by (meson "Act-Sub:3" "Hypothetical Syllogism" "≡I" "nec-imp-act" "oa-facts:2" "oa-facts:4")
3541
3542AOT_theorem "beta-C-meta": ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n, ν⇩1...ν⇩n}]↓ → ([λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n, ν⇩1...ν⇩n}]ν⇩1...ν⇩n ≡ φ{ν⇩1...ν⇩n, ν⇩1...ν⇩n})›
3543 using "lambda-predicates:2"[axiom_inst] by blast
3544
3545AOT_theorem "beta-C-cor:1": ‹(∀ν⇩1...∀ν⇩n([λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n, ν⇩1...ν⇩n}]↓)) → ∀ν⇩1...∀ν⇩n ([λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n, ν⇩1...ν⇩n}]ν⇩1...ν⇩n ≡ φ{ν⇩1...ν⇩n, ν⇩1...ν⇩n})›
3546 apply (rule "cqt-basic:14"[where 'a='a, THEN "→E"])
3547 using "beta-C-meta" "∀I" by fast
3548
3549AOT_theorem "beta-C-cor:2": ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]↓ → ∀ν⇩1...∀ν⇩n ([λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]ν⇩1...ν⇩n ≡ φ{ν⇩1...ν⇩n})›
3550 apply (rule "→I"; rule "∀I")
3551 using "beta-C-meta"[THEN "→E"] by fast
3552
3553
3554theorem "beta-C-cor:3": assumes ‹⋀ν⇩1ν⇩n. AOT_instance_of_cqt_2 (φ (AOT_term_of_var ν⇩1ν⇩n))›
3555 shows ‹[v ⊨ ∀ν⇩1...∀ν⇩n ([λμ⇩1...μ⇩n φ{ν⇩1...ν⇩n,μ⇩1...μ⇩n}]ν⇩1...ν⇩n ≡ φ{ν⇩1...ν⇩n,ν⇩1...ν⇩n})]›
3556 using "cqt:2[lambda]"[axiom_inst, OF assms] "beta-C-cor:1"[THEN "→E"] "∀I" by fast
3557
3558AOT_theorem "betaC:1:a": ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]κ⇩1...κ⇩n ❙⊢⇩□ φ{κ⇩1...κ⇩n}›
3559proof -
3560 AOT_modally_strict {
3561 AOT_assume ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]κ⇩1...κ⇩n›
3562 moreover AOT_have ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]↓› and ‹κ⇩1...κ⇩n↓›
3563 using calculation "cqt:5:a"[axiom_inst, THEN "→E"] "&E" by blast+
3564 ultimately AOT_show ‹φ{κ⇩1...κ⇩n}›
3565 using "beta-C-cor:2"[THEN "→E", THEN "∀E"(1), THEN "≡E"(1)] by blast
3566 }
3567qed
3568
3569AOT_theorem "betaC:1:b": ‹¬φ{κ⇩1...κ⇩n} ❙⊢⇩□ ¬[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]κ⇩1...κ⇩n›
3570 using "betaC:1:a" "raa-cor:3" by blast
3571
3572lemmas "β→C" = "betaC:1:a" "betaC:1:b"
3573
3574AOT_theorem "betaC:2:a": ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]↓, κ⇩1...κ⇩n↓, φ{κ⇩1...κ⇩n} ❙⊢⇩□ [λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]κ⇩1...κ⇩n›
3575proof -
3576 AOT_modally_strict {
3577 AOT_assume 1: ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]↓› and 2: ‹κ⇩1...κ⇩n↓› and 3: ‹φ{κ⇩1...κ⇩n}›
3578 AOT_hence ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]κ⇩1...κ⇩n›
3579 using "beta-C-cor:2"[THEN "→E", OF 1, THEN "∀E"(1), THEN "≡E"(2)] by blast
3580 }
3581 AOT_thus ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]↓, κ⇩1...κ⇩n↓, φ{κ⇩1...κ⇩n} ❙⊢⇩□ [λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]κ⇩1...κ⇩n›
3582 by blast
3583qed
3584
3585AOT_theorem "betaC:2:b": ‹[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]↓, κ⇩1...κ⇩n↓, ¬[λμ⇩1...μ⇩n φ{μ⇩1...μ⇩n}]κ⇩1...κ⇩n ❙⊢⇩□ ¬φ{κ⇩1...κ⇩n}›
3586 using "betaC:2:a" "raa-cor:3" by blast
3587
3588lemmas "β←C" = "betaC:2:a" "betaC:2:b"
3589
3590AOT_theorem "eta-conversion-lemma1:1": ‹Π↓ → [λx⇩1...x⇩n [Π]x⇩1...x⇩n] = Π›
3591 using "lambda-predicates:3"[axiom_inst] "∀I" "∀E"(1) "→I" by fast
3592
3593AOT_theorem "eta-conversion-lemma1:2": ‹Π↓ → [λν⇩1...ν⇩n [Π]ν⇩1...ν⇩n] = Π›
3594 using "eta-conversion-lemma1:1".
3595
3596
3597
3598text‹Note: not explicitly part of PLM.›
3599AOT_theorem id_sym: assumes ‹τ = τ'› shows ‹τ' = τ›
3600 using "rule=E"[where φ="λ τ' . «τ' = τ»", rotated 1, OF assms]
3601 "=I"(1)[OF "t=t-proper:1"[THEN "→E", OF assms]] by auto
3602declare id_sym[sym]
3603
3604text‹Note: not explicitly part of PLM.›
3605AOT_theorem id_trans: assumes ‹τ = τ'› and ‹τ' = τ''› shows ‹τ = τ''›
3606 using "rule=E" assms by blast
3607declare id_trans[trans]
3608
3609method "ηC" for Π :: ‹<'a::{AOT_Term_id_2,AOT_κs}>› = (match conclusion in "[v ⊨ τ{Π} = τ'{Π}]" for v τ τ' ⇒ ‹
3610rule "rule=E"[rotated 1, OF "eta-conversion-lemma1:2"[THEN "→E", of v "«[Π]»", symmetric]]
3611›)
3612
3618
3619
3620AOT_theorem "sub-des-lam:1": ‹[λz⇩1...z⇩n χ{z⇩1...z⇩n, ❙ιx φ{x}}]↓ & ❙ιx φ{x} = ❙ιx ψ{x} → [λz⇩1...z⇩n χ{z⇩1...z⇩n, ❙ιx φ{x}}] = [λz⇩1...z⇩n χ{z⇩1...z⇩n, ❙ιx ψ{x}}]›
3621proof(rule "→I")
3622 AOT_assume A: ‹[λz⇩1...z⇩n χ{z⇩1...z⇩n, ❙ιx φ{x}}]↓ & ❙ιx φ{x} = ❙ιx ψ{x}›
3623 AOT_show ‹[λz⇩1...z⇩n χ{z⇩1...z⇩n, ❙ιx φ{x}}] = [λz⇩1...z⇩n χ{z⇩1...z⇩n, ❙ιx ψ{x}}]›
3624 using "rule=E"[where φ="λ τ . «[λz⇩1...z⇩n χ{z⇩1...z⇩n, ❙ιx φ{x}}] = [λz⇩1...z⇩n χ{z⇩1...z⇩n, τ}]»",
3625 OF "=I"(1)[OF A[THEN "&E"(1)]], OF A[THEN "&E"(2)]]
3626 by blast
3627qed
3628
3629AOT_theorem "sub-des-lam:2": ‹❙ιx φ{x} = ❙ιx ψ{x} → χ{❙ιx φ{x}} = χ{❙ιx ψ{x}}› for χ :: ‹κ ⇒ 𝗈›
3630 using "rule=E"[where φ="λ τ . «χ{❙ιx φ{x}} = χ{τ}»", OF "=I"(1)[OF "log-prop-prop:2"]] "→I" by blast
3631
3632AOT_theorem "prop-equiv": ‹F = G ≡ ∀x (x[F] ≡ x[G])›
3633proof(rule "≡I"; rule "→I")
3634 AOT_assume ‹F = G›
3635 AOT_thus ‹∀x (x[F] ≡ x[G])›
3636 by (rule "rule=E"[rotated]) (fact "oth-class-taut:3:a"[THEN GEN])
3637next
3638 AOT_assume ‹∀x (x[F] ≡ x[G])›
3639 AOT_hence ‹x[F] ≡ x[G]› for x using "∀E" by blast
3640 AOT_hence ‹□(x[F] ≡ x[G])› for x using "en-eq:6[1]"[THEN "≡E"(1)] by blast
3641 AOT_hence ‹∀x □(x[F] ≡ x[G])› by (rule GEN)
3642 AOT_hence ‹□∀x (x[F] ≡ x[G])› using BF[THEN "→E"] by fast
3643 AOT_thus "F = G" using "p-identity-thm2:1"[THEN "≡E"(2)] by blast
3644qed
3645
3646AOT_theorem "relations:1":
3647 assumes ‹INSTANCE_OF_CQT_2(φ)›
3648 shows ‹∃F □∀x⇩1...∀x⇩n ([F]x⇩1...x⇩n ≡ φ{x⇩1...x⇩n})›
3649 apply (rule "∃I"(1)[where τ="«[λx⇩1...x⇩n φ{x⇩1...x⇩n}]»"])
3650 using "cqt:2[lambda]"[OF assms, axiom_inst] "beta-C-cor:2"[THEN "→E", THEN RN] by blast+
3651
3652AOT_theorem "relations:2":
3653 assumes ‹INSTANCE_OF_CQT_2(φ)›
3654 shows ‹∃F □∀x ([F]x ≡ φ{x})›
3655 using "relations:1" assms by blast
3656
3657AOT_theorem "block-paradox:1": ‹¬[λx ∃G (x[G] & ¬[G]x)]↓›
3658proof(rule RAA(2))
3659 let ?φ="λ τ. «∃G (τ[G] & ¬[G]τ)»"
3660 AOT_assume A: ‹[λx «?φ x»]↓›
3661 AOT_have ‹∃x (A!x & ∀F (x[F] ≡ F = [λx «?φ x»]))›
3662 using "A-objects"[axiom_inst] by fast
3663 then AOT_obtain a where ξ: ‹A!a & ∀F (a[F] ≡ F = [λx «?φ x»])›
3664 using "∃E"[rotated] by blast
3665 AOT_show ‹¬[λx ∃G (x[G] & ¬[G]x)]↓›
3666 proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3667 AOT_assume B: ‹[λx «?φ x»]a›
3668 AOT_hence ‹∃G (a[G] & ¬[G]a)› using "β→C" A by blast
3669 then AOT_obtain P where ‹a[P] & ¬[P]a› using "∃E"[rotated] by blast
3670 moreover AOT_have ‹P = [λx «?φ x»]›
3671 using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)] calculation[THEN "&E"(1)] by blast
3672 ultimately AOT_have ‹¬[λx «?φ x»]a›
3673 using "rule=E" "&E"(2) by fast
3674 AOT_thus ‹¬[λx ∃G (x[G] & ¬[G]x)]↓› using B RAA by blast
3675 next
3676 AOT_assume B: ‹¬[λx «?φ x»]a›
3677 AOT_hence ‹¬∃G (a[G] & ¬[G]a)› using "β←C" "cqt:2[const_var]"[of a, axiom_inst] A by blast
3678 AOT_hence C: ‹∀G ¬(a[G] & ¬[G]a)› using "cqt-further:4"[THEN "→E"] by blast
3679 AOT_have ‹∀G (a[G] → [G]a)›
3680 by (AOT_subst ‹a[G] → [G]a› ‹¬(a[G] & ¬[G]a)› bound: G)
3681 (auto simp: "oth-class-taut:1:a" C)
3682 AOT_hence ‹a[λx «?φ x»] → [λx «?φ x»]a› using "∀E" A by blast
3683 moreover AOT_have ‹a[λx «?φ x»]› using ξ[THEN "&E"(2), THEN "∀E"(1), OF A, THEN "≡E"(2)]
3684 using "=I"(1)[OF A] by blast
3685 ultimately AOT_show ‹¬[λx ∃G (x[G] & ¬[G]x)]↓› using B "→E" RAA by blast
3686 qed
3687qed(simp)
3688
3689AOT_theorem "block-paradox:2": ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
3690proof(rule RAA(2))
3691 AOT_assume ‹∃F ∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))›
3692 then AOT_obtain F where F_prop: ‹∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))› using "∃E"[rotated] by blast
3693 AOT_have ‹∃x (A!x & ∀G (x[G] ≡ G = F))›
3694 using "A-objects"[axiom_inst] by fast
3695 then AOT_obtain a where ξ: ‹A!a & ∀G (a[G] ≡ G = F)›
3696 using "∃E"[rotated] by blast
3697 AOT_show ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
3698 proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3699 AOT_assume B: ‹[F]a›
3700 AOT_hence ‹∃G (a[G] & ¬[G]a)› using F_prop[THEN "∀E"(2), THEN "≡E"(1)] by blast
3701 then AOT_obtain P where ‹a[P] & ¬[P]a› using "∃E"[rotated] by blast
3702 moreover AOT_have ‹P = F›
3703 using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)] calculation[THEN "&E"(1)] by blast
3704 ultimately AOT_have ‹¬[F]a›
3705 using "rule=E" "&E"(2) by fast
3706 AOT_thus ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))› using B RAA by blast
3707 next
3708 AOT_assume B: ‹¬[F]a›
3709 AOT_hence ‹¬∃G (a[G] & ¬[G]a)›
3710 using "oth-class-taut:4:b"[THEN "≡E"(1), OF F_prop[THEN "∀E"(2)[of _ _ a]], THEN "≡E"(1)] by simp
3711 AOT_hence C: ‹∀G ¬(a[G] & ¬[G]a)› using "cqt-further:4"[THEN "→E"] by blast
3712 AOT_have ‹∀G (a[G] → [G]a)›
3713 by (AOT_subst ‹a[G] → [G]a› ‹¬(a[G] & ¬[G]a)› bound: G)
3714 (auto simp: "oth-class-taut:1:a" C)
3715 AOT_hence ‹a[F] → [F]a› using "∀E" by blast
3716 moreover AOT_have ‹a[F]› using ξ[THEN "&E"(2), THEN "∀E"(2), of F, THEN "≡E"(2)]
3717 using "=I"(2) by blast
3718 ultimately AOT_show ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))› using B "→E" RAA by blast
3719 qed
3720qed(simp)
3721
3722AOT_theorem "block-paradox:3": ‹¬∀y [λz z = y]↓›
3723proof(rule RAA(2))
3724 AOT_assume θ: ‹∀y [λz z = y]↓›
3725 AOT_have ‹∃x (A!x & ∀F (x[F] ≡ ∃y(F = [λz z = y] & ¬y[F])))›
3726 using "A-objects"[axiom_inst] by force
3727 then AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y (F = [λz z = y] & ¬y[F]))›
3728 using "∃E"[rotated] by blast
3729 AOT_have ζ: ‹a[λz z = a] ≡ ∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3730 using θ[THEN "∀E"(2)] a_prop[THEN "&E"(2), THEN "∀E"(1)] by blast
3731 AOT_show ‹¬∀y [λz z = y]↓›
3732 proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3733 AOT_assume A: ‹a[λz z = a]›
3734 AOT_hence ‹∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3735 using ζ[THEN "≡E"(1)] by blast
3736 then AOT_obtain b where b_prop: ‹[λz z = a] = [λz z = b] & ¬b[λz z = a]›
3737 using "∃E"[rotated] by blast
3738 moreover AOT_have ‹a = a› by (rule "=I")
3739 moreover AOT_have ‹[λz z = a]↓› using θ "∀E" by blast
3740 moreover AOT_have ‹a↓› using "cqt:2[const_var]"[axiom_inst] .
3741 ultimately AOT_have ‹[λz z = a]a› using "β←C" by blast
3742 AOT_hence ‹[λz z = b]a› using "rule=E" b_prop[THEN "&E"(1)] by fast
3743 AOT_hence ‹a = b› using "β→C" by blast
3744 AOT_hence ‹b[λz z = a]› using A "rule=E" by fast
3745 AOT_thus ‹¬∀y [λz z = y]↓› using b_prop[THEN "&E"(2)] RAA by blast
3746 next
3747 AOT_assume A: ‹¬a[λz z = a]›
3748 AOT_hence ‹¬∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3749 using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
3750 AOT_hence ‹∀y ¬([λz z = a] = [λz z = y] & ¬y[λz z = a])›
3751 using "cqt-further:4"[THEN "→E"] by blast
3752 AOT_hence ‹¬([λz z = a] = [λz z = a] & ¬a[λz z = a])›
3753 using "∀E" by blast
3754 AOT_hence ‹[λz z = a] = [λz z = a] → a[λz z = a]›
3755 by (metis "&I" "deduction-theorem" "raa-cor:4")
3756 AOT_hence ‹a[λz z = a]› using "=I"(1) θ[THEN "∀E"(2)] "→E" by blast
3757 AOT_thus ‹¬∀y [λz z = y]↓› using A RAA by blast
3758 qed
3759qed(simp)
3760
3761AOT_theorem "block-paradox:4": ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
3762proof(rule RAA(2))
3763 AOT_assume θ: ‹∀y ∃F ∀x([F]x ≡ x = y)›
3764 AOT_have ‹∃x (A!x & ∀F (x[F] ≡ ∃z (∀y([F]y ≡ y = z) & ¬z[F])))›
3765 using "A-objects"[axiom_inst] by force
3766 then AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃z (∀y([F]y ≡ y = z) & ¬z[F]))›
3767 using "∃E"[rotated] by blast
3768 AOT_obtain F where F_prop: ‹∀x ([F]x ≡ x = a)› using θ[THEN "∀E"(2)] "∃E"[rotated] by blast
3769 AOT_have ζ: ‹a[F] ≡ ∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3770 using a_prop[THEN "&E"(2), THEN "∀E"(2)] by blast
3771 AOT_show ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
3772 proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
3773 AOT_assume A: ‹a[F]›
3774 AOT_hence ‹∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3775 using ζ[THEN "≡E"(1)] by blast
3776 then AOT_obtain b where b_prop: ‹∀y ([F]y ≡ y = b) & ¬b[F]›
3777 using "∃E"[rotated] by blast
3778 moreover AOT_have ‹[F]a› using F_prop[THEN "∀E"(2), THEN "≡E"(2)] "=I"(2) by blast
3779 ultimately AOT_have ‹a = b› using "∀E"(2) "≡E"(1) "&E" by fast
3780 AOT_hence ‹a = b› using "β→C" by blast
3781 AOT_hence ‹b[F]› using A "rule=E" by fast
3782 AOT_thus ‹¬∀y ∃F ∀x([F]x ≡ x = y)› using b_prop[THEN "&E"(2)] RAA by blast
3783 next
3784 AOT_assume A: ‹¬a[F]›
3785 AOT_hence ‹¬∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
3786 using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
3787 AOT_hence ‹∀z ¬(∀y ([F]y ≡ y = z) & ¬z[F])›
3788 using "cqt-further:4"[THEN "→E"] by blast
3789 AOT_hence ‹¬(∀y ([F]y ≡ y = a) & ¬a[F])›
3790 using "∀E" by blast
3791 AOT_hence ‹∀y ([F]y ≡ y = a) → a[F]›
3792 by (metis "&I" "deduction-theorem" "raa-cor:4")
3793 AOT_hence ‹a[F]› using F_prop "→E" by blast
3794 AOT_thus ‹¬∀y ∃F ∀x([F]x ≡ x = y)› using A RAA by blast
3795 qed
3796qed(simp)
3797
3798AOT_theorem "block-paradox:5": ‹¬∃F∀x∀y([F]xy ≡ y = x)›
3799proof(rule "raa-cor:2")
3800 AOT_assume ‹∃F∀x∀y([F]xy ≡ y = x)›
3801 then AOT_obtain F where F_prop: ‹∀x∀y([F]xy ≡ y = x)› using "∃E"[rotated] by blast
3802 {
3803 fix x
3804 AOT_have 1: ‹∀y([F]xy ≡ y = x)› using F_prop "∀E" by blast
3805 AOT_have 2: ‹[λz [F]xz]↓› by "cqt:2[lambda]"
3806 moreover AOT_have ‹∀y([λz [F]xz]y ≡ y = x)›
3807 proof(rule "∀I")
3808 fix y
3809 AOT_have ‹[λz [F]xz]y ≡ [F]xy›
3810 using "beta-C-meta"[THEN "→E"] 2 by fast
3811 also AOT_have ‹... ≡ y = x› using 1 "∀E"
3812 by fast
3813 finally AOT_show ‹[λz [F]xz]y ≡ y = x›.
3814 qed
3815 ultimately AOT_have ‹∃F∀y([F]y ≡ y = x)›
3816 using "∃I" by fast
3817 }
3818 AOT_hence ‹∀x∃F∀y([F]y ≡ y = x)›
3819 by (rule GEN)
3820 AOT_thus ‹∀x∃F∀y([F]y ≡ y = x) & ¬∀x∃F∀y([F]y ≡ y = x)›
3821 using "&I" "block-paradox:4" by blast
3822qed
3823
3824AOT_act_theorem "block-paradox2:1": ‹∀x [G]x → ¬[λx [G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3825proof(rule "→I"; rule "raa-cor:2")
3826 AOT_assume antecedant: ‹∀x [G]x›
3827 AOT_have Lemma: ‹∀x ([G]❙ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))›
3828 proof(rule GEN)
3829 fix x
3830 AOT_have A: ‹[G]❙ιy (y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3831 proof(rule "≡I"; rule "→I")
3832 AOT_assume ‹[G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))›
3833 AOT_hence ‹❙ιy (y = x & ∃H (x[H] & ¬[H]x))↓›
3834 using "cqt:5:a"[axiom_inst, THEN "→E", THEN "&E"(2)] by blast
3835 AOT_thus ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3836 using "1-exists:1"[THEN "≡E"(1)] by blast
3837 next
3838 AOT_assume A: ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3839 AOT_obtain a where a_1: ‹a = x & ∃H (x[H] & ¬[H]x)› and a_2: ‹∀z (z = x & ∃H (x[H] & ¬[H]x) → z = a)›
3840 using "uniqueness:1"[THEN "≡⇩d⇩fE", OF A] "&E" "∃E"[rotated] by blast
3841 AOT_have a_3: ‹[G]a›
3842 using antecedant "∀E" by blast
3843 AOT_show ‹[G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))›
3844 apply (rule "russell-axiom[exe,1].russell-axiom"[THEN "≡E"(2)])
3845 apply (rule "∃I"(2))
3846 using a_1 a_2 a_3 "&I" by blast
3847 qed
3848 also AOT_have B: ‹... ≡ ∃H (x[H] & ¬[H]x)›
3849 proof (rule "≡I"; rule "→I")
3850 AOT_assume A: ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3851 AOT_obtain a where ‹a = x & ∃H (x[H] & ¬[H]x)›
3852 using "uniqueness:1"[THEN "≡⇩d⇩fE", OF A] "&E" "∃E"[rotated] by blast
3853 AOT_thus ‹∃H (x[H] & ¬[H]x)› using "&E" by blast
3854 next
3855 AOT_assume ‹∃H (x[H] & ¬[H]x)›
3856 AOT_hence ‹x = x & ∃H (x[H] & ¬[H]x)›
3857 using "id-eq:1" "&I" by blast
3858 moreover AOT_have ‹∀z (z = x & ∃H (x[H] & ¬[H]x) → z = x)›
3859 by (simp add: "Conjunction Simplification"(1) "universal-cor")
3860 ultimately AOT_show ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
3861 using "uniqueness:1"[THEN "≡⇩d⇩fI"] "&I" "∃I"(2) by fast
3862 qed
3863 finally AOT_show ‹([G]❙ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))› .
3864 qed
3865
3866 AOT_assume A: ‹[λx [G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3867 AOT_have θ: ‹∀x ([λx [G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))]x ≡ [G]❙ιy(y = x & ∃H (x[H] & ¬[H]x)))›
3868 using "beta-C-meta"[THEN "→E", OF A] "∀I" by fast
3869 AOT_have ‹∀x ([λx [G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))]x ≡ ∃H (x[H] & ¬[H]x))›
3870 using θ Lemma "cqt-basic:10"[THEN "→E"] "&I" by fast
3871 AOT_hence ‹∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x))›
3872 using "∃I"(1) A by fast
3873 AOT_thus ‹(∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x))) & (¬∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x)))›
3874 using "block-paradox:2" "&I" by blast
3875qed
3876
3877AOT_act_theorem "block-paradox2:2": ‹∃G ¬[λx [G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3878proof(rule "∃I"(1))
3879 AOT_have 0: ‹[λx ∀p (p →p)]↓›
3880 by "cqt:2[lambda]"
3881 moreover AOT_have ‹∀x [λx ∀p (p →p)]x›
3882 apply (rule GEN)
3883 apply (rule "beta-C-cor:2"[THEN "→E", OF 0, THEN "∀E"(2), THEN "≡E"(2)])
3884 using "if-p-then-p" GEN by fast
3885 moreover AOT_have ‹∀G (∀x [G]x → ¬[λx [G]❙ιy (y = x & ∃H (x[H] & ¬[H]x))]↓)›
3886 using "block-paradox2:1" "∀I" by fast
3887 ultimately AOT_show ‹¬[λx [λx ∀p (p →p)]❙ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
3888 using "∀E"(1) "→E" by blast
3889qed("cqt:2[lambda]")
3890
3891AOT_theorem propositions: ‹∃p □(p ≡ φ)›
3892proof(rule "∃I"(1))
3893 AOT_show ‹□(φ ≡ φ)›
3894 by (simp add: RN "oth-class-taut:3:a")
3895next
3896 AOT_show ‹φ↓›
3897 by (simp add: "log-prop-prop:2")
3898qed
3899
3900AOT_theorem "pos-not-equiv-ne:1": ‹(◇¬∀x⇩1...∀x⇩n ([F]x⇩1...x⇩n ≡ [G]x⇩1...x⇩n)) → F ≠ G›
3901proof (rule "→I")
3902 AOT_assume ‹◇¬∀x⇩1...∀x⇩n ([F]x⇩1...x⇩n ≡ [G]x⇩1...x⇩n)›
3903 AOT_hence ‹¬□∀x⇩1...∀x⇩n ([F]x⇩1...x⇩n ≡ [G]x⇩1...x⇩n)›
3904 using "KBasic:11"[THEN "≡E"(2)] by blast
3905 AOT_hence ‹¬(F = G)›
3906 using "id-rel-nec-equiv:1" "modus-tollens:1" by blast
3907 AOT_thus ‹F ≠ G›
3908 using "=-infix"[THEN "≡⇩d⇩fI"] by blast
3909qed
3910
3911AOT_theorem "pos-not-equiv-ne:2": ‹(◇¬(φ{F} ≡ φ{G})) → F ≠ G›
3912proof (rule "→I")
3913 AOT_modally_strict {
3914 AOT_have ‹¬(φ{F} ≡ φ{G}) → ¬(F = G)›
3915 proof (rule "→I"; rule "raa-cor:2")
3916 AOT_assume 1: ‹F = G›
3917 AOT_hence ‹φ{F} → φ{G}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3918 moreover {
3919 AOT_have ‹G = F› using 1 id_sym by blast
3920 AOT_hence ‹φ{G} → φ{F}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3921 }
3922 ultimately AOT_have ‹φ{F} ≡ φ{G}› using "≡I" by blast
3923 moreover AOT_assume ‹¬(φ{F} ≡ φ{G})›
3924 ultimately AOT_show ‹(φ{F} ≡ φ{G}) & ¬(φ{F} ≡ φ{G})›
3925 using "&I" by blast
3926 qed
3927 }
3928 AOT_hence ‹◇¬(φ{F} ≡ φ{G}) → ◇¬(F = G)›
3929 using "RM:2[prem]" by blast
3930 moreover AOT_assume ‹◇¬(φ{F} ≡ φ{G})›
3931 ultimately AOT_have 0: ‹◇¬(F = G)› using "→E" by blast
3932 AOT_have ‹◇(F ≠ G)›
3933 by (AOT_subst ‹F ≠ G› ‹¬(F = G)›)
3934 (auto simp: "=-infix" "≡Df" 0)
3935 AOT_thus ‹F ≠ G›
3936 using "id-nec2:3"[THEN "→E"] by blast
3937qed
3938
3939AOT_theorem "pos-not-equiv-ne:2[zero]": ‹(◇¬(φ{p} ≡ φ{q})) → p ≠ q›
3940proof (rule "→I")
3941 AOT_modally_strict {
3942 AOT_have ‹¬(φ{p} ≡ φ{q}) → ¬(p = q)›
3943 proof (rule "→I"; rule "raa-cor:2")
3944 AOT_assume 1: ‹p = q›
3945 AOT_hence ‹φ{p} → φ{q}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3946 moreover {
3947 AOT_have ‹q = p› using 1 id_sym by blast
3948 AOT_hence ‹φ{q} → φ{p}› using "l-identity"[axiom_inst, THEN "→E"] by blast
3949 }
3950 ultimately AOT_have ‹φ{p} ≡ φ{q}› using "≡I" by blast
3951 moreover AOT_assume ‹¬(φ{p} ≡ φ{q})›
3952 ultimately AOT_show ‹(φ{p} ≡ φ{q}) & ¬(φ{p} ≡ φ{q})›
3953 using "&I" by blast
3954 qed
3955 }
3956 AOT_hence ‹◇¬(φ{p} ≡ φ{q}) → ◇¬(p = q)›
3957 using "RM:2[prem]" by blast
3958 moreover AOT_assume ‹◇¬(φ{p} ≡ φ{q})›
3959 ultimately AOT_have 0: ‹◇¬(p = q)› using "→E" by blast
3960 AOT_have ‹◇(p ≠ q)›
3961 by (AOT_subst ‹p ≠ q› ‹¬(p = q)›)
3962 (auto simp: 0 "=-infix" "≡Df")
3963 AOT_thus ‹p ≠ q›
3964 using "id-nec2:3"[THEN "→E"] by blast
3965qed
3966
3967AOT_theorem "pos-not-equiv-ne:3": ‹(¬∀x⇩1...∀x⇩n ([F]x⇩1...x⇩n ≡ [G]x⇩1...x⇩n)) → F ≠ G›
3968 using "→I" "pos-not-equiv-ne:1"[THEN "→E"] "T◇"[THEN "→E"] by blast
3969
3970AOT_theorem "pos-not-equiv-ne:4": ‹(¬(φ{F} ≡ φ{G})) → F ≠ G›
3971 using "→I" "pos-not-equiv-ne:2"[THEN "→E"] "T◇"[THEN "→E"] by blast
3972
3973AOT_theorem "pos-not-equiv-ne:4[zero]": ‹(¬(φ{p} ≡ φ{q})) → p ≠ q›
3974 using "→I" "pos-not-equiv-ne:2[zero]"[THEN "→E"] "T◇"[THEN "→E"] by blast
3975
3976AOT_define relation_negation :: "Π ⇒ Π" ("_⇧-")
3977 "df-relation-negation": "[F]⇧- =⇩d⇩f [λx⇩1...x⇩n ¬[F]x⇩1...x⇩n]"
3978
3979nonterminal φneg
3980syntax "" :: "φneg ⇒ τ" ("_")
3981syntax "" :: "φneg ⇒ φ" ("'(_')")
3982
3983AOT_define relation_negation_0 :: ‹φ ⇒ φneg› ("'(_')⇧-")
3984 "df-relation-negation[zero]": "(p)⇧- =⇩d⇩f [λ ¬p]"
3985
3986AOT_theorem "rel-neg-T:1": ‹[λx⇩1...x⇩n ¬[Π]x⇩1...x⇩n]↓›
3987 by "cqt:2[lambda]"
3988
3989AOT_theorem "rel-neg-T:1[zero]": ‹[λ ¬φ]↓›
3990 using "cqt:2[lambda0]"[axiom_inst] by blast
3991
3992AOT_theorem "rel-neg-T:2": ‹[Π]⇧- = [λx⇩1...x⇩n ¬[Π]x⇩1...x⇩n]›
3993 using "=I"(1)[OF "rel-neg-T:1"]
3994 by (rule "=⇩d⇩fI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"])
3995
3996AOT_theorem "rel-neg-T:2[zero]": ‹(φ)⇧- = [λ ¬φ]›
3997 using "=I"(1)[OF "rel-neg-T:1[zero]"]
3998 by (rule "=⇩d⇩fI"(1)[OF "df-relation-negation[zero]", OF "rel-neg-T:1[zero]"])
3999
4000AOT_theorem "rel-neg-T:3": ‹[Π]⇧-↓›
4001 using "=⇩d⇩fI"(1)[OF "df-relation-negation", OF "rel-neg-T:1"] "rel-neg-T:1" by blast
4002
4003AOT_theorem "rel-neg-T:3[zero]": ‹(φ)⇧-↓›
4004 using "log-prop-prop:2" by blast
4005
4006
4007
4008AOT_theorem "thm-relation-negation:1": ‹[F]⇧-x⇩1...x⇩n ≡ ¬[F]x⇩1...x⇩n›
4009proof -
4010 AOT_have ‹[F]⇧-x⇩1...x⇩n ≡ [λx⇩1...x⇩n ¬[F]x⇩1...x⇩n]x⇩1...x⇩n›
4011 using "rule=E"[rotated, OF "rel-neg-T:2"] "rule=E"[rotated, OF "rel-neg-T:2"[THEN id_sym]]
4012 "→I" "≡I" by fast
4013 also AOT_have ‹... ≡ ¬[F]x⇩1...x⇩n›
4014 using "beta-C-meta"[THEN "→E", OF "rel-neg-T:1"] by fast
4015 finally show ?thesis.
4016qed
4017
4018AOT_theorem "thm-relation-negation:2": ‹¬[F]⇧-x⇩1...x⇩n ≡ [F]x⇩1...x⇩n›
4019 apply (AOT_subst ‹[F]x⇩1...x⇩n› ‹¬¬[F]x⇩1...x⇩n›)
4020 apply (simp add: "oth-class-taut:3:b")
4021 apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4022 using "thm-relation-negation:1".
4023
4024AOT_theorem "thm-relation-negation:3": ‹((p)⇧-) ≡ ¬p›
4025proof -
4026 AOT_have ‹(p)⇧- = [λ ¬p]› using "rel-neg-T:2[zero]" by blast
4027 AOT_hence ‹((p)⇧-) ≡ [λ ¬p]›
4028 using "df-relation-negation[zero]" "log-prop-prop:2" "oth-class-taut:3:a" "rule-id-df:2:a" by blast
4029 also AOT_have ‹[λ ¬p] ≡ ¬p›
4030 by (simp add: "propositions-lemma:2")
4031 finally show ?thesis.
4032qed
4033
4034AOT_theorem "thm-relation-negation:4": ‹(¬((p)⇧-)) ≡ p›
4035 using "thm-relation-negation:3"[THEN "≡E"(1)]
4036 "thm-relation-negation:3"[THEN "≡E"(2)]
4037 "≡I" "→I" RAA by metis
4038
4039AOT_theorem "thm-relation-negation:5": ‹[F] ≠ [F]⇧-›
4040proof -
4041 AOT_have ‹¬([F] = [F]⇧-)›
4042 proof (rule RAA(2))
4043 AOT_show ‹[F]x⇩1...x⇩n → [F]x⇩1...x⇩n› for x⇩1x⇩n
4044 using "if-p-then-p".
4045 next
4046 AOT_assume ‹[F] = [F]⇧-›
4047 AOT_hence ‹[F]⇧- = [F]› using id_sym by blast
4048 AOT_hence ‹[F]x⇩1...x⇩n ≡ ¬[F]x⇩1...x⇩n› for x⇩1x⇩n
4049 using "rule=E" "thm-relation-negation:1" by fast
4050 AOT_thus ‹¬([F]x⇩1...x⇩n → [F]x⇩1...x⇩n)› for x⇩1x⇩n
4051 using "≡E" RAA by metis
4052 qed
4053 thus ?thesis
4054 using "≡⇩d⇩fI" "=-infix" by blast
4055qed
4056
4057AOT_theorem "thm-relation-negation:6": ‹p ≠ (p)⇧-›
4058proof -
4059 AOT_have ‹¬(p = (p)⇧-)›
4060 proof (rule RAA(2))
4061 AOT_show ‹p → p›
4062 using "if-p-then-p".
4063 next
4064 AOT_assume ‹p = (p)⇧-›
4065 AOT_hence ‹(p)⇧- = p› using id_sym by blast
4066 AOT_hence ‹p ≡ ¬p›
4067 using "rule=E" "thm-relation-negation:3" by fast
4068 AOT_thus ‹¬(p → p)›
4069 using "≡E" RAA by metis
4070 qed
4071 thus ?thesis
4072 using "≡⇩d⇩fI" "=-infix" by blast
4073qed
4074
4075AOT_theorem "thm-relation-negation:7": ‹(p)⇧- = (¬p)›
4076 apply (rule "df-relation-negation[zero]"[THEN "=⇩d⇩fE"(1)])
4077 using "cqt:2[lambda0]"[axiom_inst] "rel-neg-T:2[zero]" "propositions-lemma:1" id_trans by blast+
4078
4079AOT_theorem "thm-relation-negation:8": ‹p = q → (¬p) = (¬q)›
4080proof(rule "→I")
4081 AOT_assume ‹p = q›
4082 moreover AOT_have ‹(¬p)↓› using "log-prop-prop:2".
4083 moreover AOT_have ‹(¬p) = (¬p)› using calculation(2) "=I" by blast
4084 ultimately AOT_show ‹(¬p) = (¬q)›
4085 using "rule=E" by fast
4086qed
4087
4088AOT_theorem "thm-relation-negation:9": ‹p = q → (p)⇧- = (q)⇧-›
4089proof(rule "→I")
4090 AOT_assume ‹p = q›
4091 AOT_hence ‹(¬p) = (¬q)› using "thm-relation-negation:8" "→E" by blast
4092 AOT_thus ‹(p)⇧- = (q)⇧-›
4093 using "thm-relation-negation:7" id_sym id_trans by metis
4094qed
4095
4096AOT_define Necessary :: ‹Π ⇒ φ› ("Necessary'(_')")
4097 "contingent-properties:1": ‹Necessary([F]) ≡⇩d⇩f □∀x⇩1...∀x⇩n [F]x⇩1...x⇩n›
4098
4099AOT_define Necessary0 :: ‹φ ⇒ φ› ("Necessary0'(_')")
4100 "contingent-properties:1[zero]": ‹Necessary0(p) ≡⇩d⇩f □p›
4101
4102AOT_define Impossible :: ‹Π ⇒ φ› ("Impossible'(_')")
4103 "contingent-properties:2": ‹Impossible([F]) ≡⇩d⇩f F↓ & □∀x⇩1...∀x⇩n ¬[F]x⇩1...x⇩n›
4104
4105AOT_define Impossible0 :: ‹φ ⇒ φ› ("Impossible0'(_')")
4106 "contingent-properties:2[zero]": ‹Impossible0(p) ≡⇩d⇩f □¬p›
4107
4108AOT_define NonContingent :: ‹Π ⇒ φ› ("NonContingent'(_')")
4109 "contingent-properties:3": ‹NonContingent([F]) ≡⇩d⇩f Necessary([F]) ∨ Impossible([F])›
4110
4111AOT_define NonContingent0 :: ‹φ ⇒ φ› ("NonContingent0'(_')")
4112 "contingent-properties:3[zero]": ‹NonContingent0(p) ≡⇩d⇩f Necessary0(p) ∨ Impossible0(p)›
4113
4114AOT_define Contingent :: ‹Π ⇒ φ› ("Contingent'(_')")
4115 "contingent-properties:4": ‹Contingent([F]) ≡⇩d⇩f F↓ & ¬(Necessary([F]) ∨ Impossible([F]))›
4116
4117AOT_define Contingent0 :: ‹φ ⇒ φ› ("Contingent0'(_')")
4118 "contingent-properties:4[zero]": ‹Contingent0(p) ≡⇩d⇩f ¬(Necessary0(p) ∨ Impossible0(p))›
4119
4120
4121AOT_theorem "thm-cont-prop:1": ‹NonContingent([F]) ≡ NonContingent([F]⇧-)›
4122proof (rule "≡I"; rule "→I")
4123 AOT_assume ‹NonContingent([F])›
4124 AOT_hence ‹Necessary([F]) ∨ Impossible([F])›
4125 using "≡⇩d⇩fE"[OF "contingent-properties:3"] by blast
4126 moreover {
4127 AOT_assume ‹Necessary([F])›
4128 AOT_hence ‹□(∀x⇩1...∀x⇩n [F]x⇩1...x⇩n)›
4129 using "≡⇩d⇩fE"[OF "contingent-properties:1"] by blast
4130 moreover AOT_modally_strict {
4131 AOT_assume ‹∀x⇩1...∀x⇩n [F]x⇩1...x⇩n›
4132 AOT_hence ‹[F]x⇩1...x⇩n› for x⇩1x⇩n using "∀E" by blast
4133 AOT_hence ‹¬[F]⇧-x⇩1...x⇩n› for x⇩1x⇩n
4134 by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:2" "≡E"(1))
4135 AOT_hence ‹∀x⇩1...∀x⇩n ¬[F]⇧-x⇩1...x⇩n› using "∀I" by fast
4136 }
4137 ultimately AOT_have ‹□(∀x⇩1...∀x⇩n ¬[F]⇧-x⇩1...x⇩n)›
4138 using "RN[prem]"[where Γ="{«∀x⇩1...∀x⇩n [F]x⇩1...x⇩n»}", simplified] by blast
4139 AOT_hence ‹Impossible([F]⇧-)›
4140 using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "rel-neg-T:3", THEN "≡E"(2)]
4141 by blast
4142 }
4143 moreover {
4144 AOT_assume ‹Impossible([F])›
4145 AOT_hence ‹□(∀x⇩1...∀x⇩n ¬[F]x⇩1...x⇩n)›
4146 using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)]
4147 by blast
4148 moreover AOT_modally_strict {
4149 AOT_assume ‹∀x⇩1...∀x⇩n ¬[F]x⇩1...x⇩n›
4150 AOT_hence ‹¬[F]x⇩1...x⇩n› for x⇩1x⇩n using "∀E" by blast
4151 AOT_hence ‹[F]⇧-x⇩1...x⇩n› for x⇩1x⇩n
4152 by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:1" "≡E"(1))
4153 AOT_hence ‹∀x⇩1...∀x⇩n [F]⇧-x⇩1...x⇩n› using "∀I" by fast
4154 }
4155 ultimately AOT_have ‹□(∀x⇩1...∀x⇩n [F]⇧-x⇩1...x⇩n)›
4156 using "RN[prem]"[where Γ="{«∀x⇩1...∀x⇩n ¬[F]x⇩1...x⇩n»}"] by blast
4157 AOT_hence ‹Necessary([F]⇧-)›
4158 using "≡⇩d⇩fI"[OF "contingent-properties:1"] by blast
4159 }
4160 ultimately AOT_have ‹Necessary([F]⇧-) ∨ Impossible([F]⇧-)›
4161 using "∨E"(1) "∨I" "→I" by metis
4162 AOT_thus ‹NonContingent([F]⇧-)›
4163 using "≡⇩d⇩fI"[OF "contingent-properties:3"] by blast
4164next
4165 AOT_assume ‹NonContingent([F]⇧-)›
4166 AOT_hence ‹Necessary([F]⇧-) ∨ Impossible([F]⇧-)›
4167 using "≡⇩d⇩fE"[OF "contingent-properties:3"] by blast
4168 moreover {
4169 AOT_assume ‹Necessary([F]⇧-)›
4170 AOT_hence ‹□(∀x⇩1...∀x⇩n [F]⇧-x⇩1...x⇩n)›
4171 using "≡⇩d⇩fE"[OF "contingent-properties:1"] by blast
4172 moreover AOT_modally_strict {
4173 AOT_assume ‹∀x⇩1...∀x⇩n [F]⇧-x⇩1...x⇩n›
4174 AOT_hence ‹[F]⇧-x⇩1...x⇩n› for x⇩1x⇩n using "∀E" by blast
4175 AOT_hence ‹¬[F]x⇩1...x⇩n› for x⇩1x⇩n
4176 by (meson "≡E"(6) "oth-class-taut:3:a" "thm-relation-negation:1" "≡E"(2))
4177 AOT_hence ‹∀x⇩1...∀x⇩n ¬[F]x⇩1...x⇩n› using "∀I" by fast
4178 }
4179 ultimately AOT_have ‹□∀x⇩1...∀x⇩n ¬[F]x⇩1...x⇩n›
4180 using "RN[prem]"[where Γ="{«∀x⇩1...∀x⇩n [F]⇧-x⇩1...x⇩n»}"] by blast
4181 AOT_hence ‹Impossible([F])›
4182 using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(2)]
4183 by blast
4184 }
4185 moreover {
4186 AOT_assume ‹Impossible([F]⇧-)›
4187 AOT_hence ‹□(∀x⇩1...∀x⇩n ¬[F]⇧-x⇩1...x⇩n)›
4188 using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1), OF "rel-neg-T:3", THEN "≡E"(1)]
4189 by blast
4190 moreover AOT_modally_strict {
4191 AOT_assume ‹∀x⇩1...∀x⇩n ¬[F]⇧-x⇩1...x⇩n›
4192 AOT_hence ‹¬[F]⇧-x⇩1...x⇩n› for x⇩1x⇩n using "∀E" by blast
4193 AOT_hence ‹[F]x⇩1...x⇩n› for x⇩1x⇩n
4194 using "thm-relation-negation:1"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4195 "useful-tautologies:1"[THEN "→E"] by blast
4196 AOT_hence ‹∀x⇩1...∀x⇩n [F]x⇩1...x⇩n› using "∀I" by fast
4197 }
4198 ultimately AOT_have ‹□(∀x⇩1...∀x⇩n [F]x⇩1...x⇩n)›
4199 using "RN[prem]"[where Γ="{«∀x⇩1...∀x⇩n ¬[F]⇧-x⇩1...x⇩n»}"] by blast
4200 AOT_hence ‹Necessary([F])›
4201 using "≡⇩d⇩fI"[OF "contingent-properties:1"] by blast
4202 }
4203 ultimately AOT_have ‹Necessary([F]) ∨ Impossible([F])›
4204 using "∨E"(1) "∨I" "→I" by metis
4205 AOT_thus ‹NonContingent([F])›
4206 using "≡⇩d⇩fI"[OF "contingent-properties:3"] by blast
4207qed
4208
4209AOT_theorem "thm-cont-prop:2": ‹Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4210proof -
4211 AOT_have ‹Contingent([F]) ≡ ¬(Necessary([F]) ∨ Impossible([F]))›
4212 using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst]]
4213 by blast
4214 also AOT_have ‹... ≡ ¬Necessary([F]) & ¬Impossible([F])›
4215 using "oth-class-taut:5:d" by fastforce
4216 also AOT_have ‹... ≡ ¬Impossible([F]) & ¬Necessary([F])›
4217 by (simp add: "Commutativity of &")
4218 also AOT_have ‹... ≡ ◇∃x [F]x & ¬Necessary([F])›
4219 proof (rule "oth-class-taut:4:e"[THEN "→E"])
4220 AOT_have ‹¬Impossible([F]) ≡ ¬□¬ ∃x [F]x›
4221 apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4222 apply (AOT_subst ‹∃x [F]x› ‹¬ ∀x ¬[F]x›)
4223 apply (simp add: "conventions:4" "≡Df")
4224 apply (AOT_subst (reverse) ‹¬¬∀x ¬[F]x› ‹∀x ¬[F]x›)
4225 apply (simp add: "oth-class-taut:3:b")
4226 using "contingent-properties:2"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst]] by blast
4227 also AOT_have ‹... ≡ ◇∃x [F]x›
4228 using "conventions:5"[THEN "≡Df", symmetric] by blast
4229 finally AOT_show ‹¬Impossible([F]) ≡ ◇∃x [F]x› .
4230 qed
4231 also AOT_have ‹... ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4232 proof (rule "oth-class-taut:4:f"[THEN "→E"])
4233 AOT_have ‹¬Necessary([F]) ≡ ¬□¬∃x ¬[F]x›
4234 apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
4235 apply (AOT_subst ‹∃x ¬[F]x› ‹¬ ∀x ¬¬[F]x›)
4236 apply (simp add: "conventions:4" "≡Df")
4237 apply (AOT_subst (reverse) ‹¬¬[F]x› ‹[F]x› bound: x)
4238 apply (simp add: "oth-class-taut:3:b")
4239 apply (AOT_subst (reverse) ‹¬¬∀x [F]x› ‹∀x [F]x›)
4240 by (auto simp: "oth-class-taut:3:b" "contingent-properties:1" "≡Df")
4241 also AOT_have ‹... ≡ ◇∃x ¬[F]x›
4242 using "conventions:5"[THEN "≡Df", symmetric] by blast
4243 finally AOT_show ‹¬Necessary([F]) ≡ ◇∃x ¬[F]x›.
4244 qed
4245 finally show ?thesis.
4246qed
4247
4248AOT_theorem "thm-cont-prop:3": ‹Contingent([F]) ≡ Contingent([F]⇧-)› for F::‹<κ> AOT_var›
4249proof -
4250 {
4251 fix Π :: ‹<κ>›
4252 AOT_assume ‹Π↓›
4253 moreover AOT_have ‹∀F (Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x)›
4254 using "thm-cont-prop:2" GEN by fast
4255 ultimately AOT_have ‹Contingent([Π]) ≡ ◇∃x [Π]x & ◇∃x ¬[Π]x›
4256 using "thm-cont-prop:2" "∀E" by fast
4257 } note 1 = this
4258 AOT_have ‹Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x›
4259 using "thm-cont-prop:2" by blast
4260 also AOT_have ‹... ≡ ◇∃x ¬[F]x & ◇∃x [F]x›
4261 by (simp add: "Commutativity of &")
4262 also AOT_have ‹... ≡ ◇∃x [F]⇧-x & ◇∃x [F]x›
4263 by (AOT_subst ‹[F]⇧-x› ‹¬[F]x› bound: x)
4264 (auto simp: "thm-relation-negation:1" "oth-class-taut:3:a")
4265 also AOT_have ‹... ≡ ◇∃x [F]⇧-x & ◇∃x ¬[F]⇧-x›
4266 by (AOT_subst (reverse) ‹[F]x› ‹¬[F]⇧-x› bound: x)
4267 (auto simp: "thm-relation-negation:2" "oth-class-taut:3:a")
4268 also AOT_have ‹... ≡ Contingent([F]⇧-)›
4269 using 1[OF "rel-neg-T:3", symmetric] by blast
4270 finally show ?thesis.
4271qed
4272
4273AOT_define concrete_if_concrete :: ‹Π› ("L") L_def: ‹L =⇩d⇩f [λx E!x → E!x]›
4274
4275AOT_theorem "thm-noncont-e-e:1": ‹Necessary(L)›
4276proof -
4277 AOT_modally_strict {
4278 fix x
4279 AOT_have ‹[λx E!x → E!x]↓› by "cqt:2[lambda]"
4280 moreover AOT_have ‹x↓› using "cqt:2[const_var]"[axiom_inst] by blast
4281 moreover AOT_have ‹E!x → E!x› using "if-p-then-p" by blast
4282 ultimately AOT_have ‹[λx E!x → E!x]x›
4283 using "β←C" by blast
4284 }
4285 AOT_hence 0: ‹□∀x [λx E!x → E!x]x›
4286 using RN GEN by blast
4287 show ?thesis
4288 apply (rule "=⇩d⇩fI"(2)[OF L_def])
4289 apply "cqt:2[lambda]"
4290 by (rule "contingent-properties:1"[THEN "≡⇩d⇩fI", OF 0])
4291qed
4292
4293AOT_theorem "thm-noncont-e-e:2": ‹Impossible([L]⇧-)›
4294proof -
4295 AOT_modally_strict {
4296 fix x
4297
4298 AOT_have 0: ‹∀F (¬[F]⇧-x ≡ [F]x)›
4299 using "thm-relation-negation:2" GEN by fast
4300 AOT_have ‹¬[λx E!x → E!x]⇧-x ≡ [λx E!x → E!x]x›
4301 by (rule 0[THEN "∀E"(1)]) "cqt:2[lambda]"
4302 moreover {
4303 AOT_have ‹[λx E!x → E!x]↓› by "cqt:2[lambda]"
4304 moreover AOT_have ‹x↓› using "cqt:2[const_var]"[axiom_inst] by blast
4305 moreover AOT_have ‹E!x → E!x› using "if-p-then-p" by blast
4306 ultimately AOT_have ‹[λx E!x → E!x]x›
4307 using "β←C" by blast
4308 }
4309 ultimately AOT_have ‹¬[λx E!x → E!x]⇧-x›
4310 using "≡E" by blast
4311 }
4312 AOT_hence 0: ‹□∀x ¬[λx E!x → E!x]⇧-x›
4313 using RN GEN by fast
4314 show ?thesis
4315 apply (rule "=⇩d⇩fI"(2)[OF L_def])
4316 apply "cqt:2[lambda]"
4317 apply (rule "contingent-properties:2"[THEN "≡⇩d⇩fI"]; rule "&I")
4318 using "rel-neg-T:3"
4319 apply blast
4320 using 0
4321 by blast
4322qed
4323
4324AOT_theorem "thm-noncont-e-e:3": ‹NonContingent(L)›
4325 using "thm-noncont-e-e:1"
4326 by (rule "contingent-properties:3"[THEN "≡⇩d⇩fI", OF "∨I"(1)])
4327
4328AOT_theorem "thm-noncont-e-e:4": ‹NonContingent([L]⇧-)›
4329proof -
4330 AOT_have 0: ‹∀F (NonContingent([F]) ≡ NonContingent([F]⇧-))›
4331 using "thm-cont-prop:1" "∀I" by fast
4332 moreover AOT_have 1: ‹L↓›
4333 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
4334 AOT_show ‹NonContingent([L]⇧-)›
4335 using "∀E"(1)[OF 0, OF 1, THEN "≡E"(1), OF "thm-noncont-e-e:3"] by blast
4336qed
4337
4338AOT_theorem "thm-noncont-e-e:5": ‹∃F ∃G (F ≠ «G::<κ>» & NonContingent([F]) & NonContingent([G]))›
4339proof (rule "∃I")+
4340 {
4341 AOT_have ‹∀F [F] ≠ [F]⇧-› using "thm-relation-negation:5" GEN by fast
4342 moreover AOT_have ‹L↓›
4343 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
4344 ultimately AOT_have ‹L ≠ [L]⇧-› using "∀E" by blast
4345 }
4346 AOT_thus ‹L ≠ [L]⇧- & NonContingent(L) & NonContingent([L]⇧-)›
4347 using "thm-noncont-e-e:3" "thm-noncont-e-e:4" "&I" by metis
4348next
4349 AOT_show ‹[L]⇧-↓›
4350 using "rel-neg-T:3" by blast
4351next
4352 AOT_show ‹L↓›
4353 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
4354qed
4355
4356AOT_theorem "lem-cont-e:1": ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x)›
4357proof -
4358 AOT_have ‹◇∃x ([F]x & ◇¬[F]x) ≡ ∃x ◇([F]x & ◇¬[F]x)›
4359 using "BF◇" "CBF◇" "≡I" by blast
4360 also AOT_have ‹… ≡ ∃x (◇[F]x & ◇¬[F]x)›
4361 by (AOT_subst ‹◇([F]x & ◇¬[F]x)› ‹◇[F]x & ◇¬[F]x› bound: x)
4362 (auto simp: "S5Basic:11" "cqt-further:7")
4363 also AOT_have ‹… ≡ ∃x (◇¬[F]x & ◇[F]x)›
4364 by (AOT_subst ‹◇¬[F]x & ◇[F]x› ‹◇[F]x & ◇¬[F]x› bound: x)
4365 (auto simp: "Commutativity of &" "cqt-further:7")
4366 also AOT_have ‹… ≡ ∃x ◇(¬[F]x & ◇[F]x)›
4367 by (AOT_subst ‹◇(¬[F]x & ◇[F]x)› ‹◇¬[F]x & ◇[F]x› bound: x)
4368 (auto simp: "S5Basic:11" "oth-class-taut:3:a")
4369 also AOT_have ‹… ≡ ◇∃x (¬[F]x & ◇[F]x)›
4370 using "BF◇" "CBF◇" "≡I" by fast
4371 finally show ?thesis.
4372qed
4373
4374AOT_theorem "lem-cont-e:2": ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x ([F]⇧-x & ◇¬[F]⇧-x)›
4375proof -
4376 AOT_have ‹◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x)›
4377 using "lem-cont-e:1".
4378 also AOT_have ‹… ≡ ◇∃x ([F]⇧-x & ◇¬[F]⇧-x)›
4379 apply (AOT_subst ‹¬[F]⇧-x› ‹[F]x› bound: x)
4380 apply (simp add: "thm-relation-negation:2")
4381 apply (AOT_subst ‹[F]⇧-x› ‹¬[F]x› bound: x)
4382 apply (simp add: "thm-relation-negation:1")
4383 by (simp add: "oth-class-taut:3:a")
4384 finally show ?thesis.
4385qed
4386
4387AOT_theorem "thm-cont-e:1": ‹◇∃x (E!x & ◇¬E!x)›
4388proof (rule "CBF◇"[THEN "→E"])
4389 AOT_have ‹∃x ◇(E!x & ¬❙𝒜E!x)› using "qml:4"[axiom_inst] "BF◇"[THEN "→E"] by blast
4390 then AOT_obtain a where ‹◇(E!a & ¬❙𝒜E!a)› using "∃E"[rotated] by blast
4391 AOT_hence θ: ‹◇E!a & ◇¬❙𝒜E!a›
4392 using "KBasic2:3"[THEN "→E"] by blast
4393 AOT_have ξ: ‹◇E!a & ◇❙𝒜¬E!a›
4394 by (AOT_subst ‹❙𝒜¬E!a› ‹¬❙𝒜E!a›)
4395 (auto simp: "logic-actual-nec:1"[axiom_inst] θ)
4396 AOT_have ζ: ‹◇E!a & ❙𝒜¬E!a›
4397 by (AOT_subst ‹❙𝒜¬E!a› ‹◇❙𝒜¬E!a›)
4398 (auto simp add: "Act-Sub:4" ξ)
4399 AOT_hence ‹◇E!a & ◇¬E!a›
4400 using "&E" "&I" "Act-Sub:3"[THEN "→E"] by blast
4401 AOT_hence ‹◇(E!a & ◇¬E!a)› using "S5Basic:11"[THEN "≡E"(2)] by simp
4402 AOT_thus ‹∃x ◇(E!x & ◇¬E!x)› using "∃I"(2) by fast
4403qed
4404
4405AOT_theorem "thm-cont-e:2": ‹◇∃x (¬E!x & ◇E!x)›
4406proof -
4407 AOT_have ‹∀F (◇∃x ([F]x & ◇¬[F]x) ≡ ◇∃x (¬[F]x & ◇[F]x))›
4408 using "lem-cont-e:1" GEN by fast
4409 AOT_hence ‹(◇∃x (E!x & ◇¬E!x) ≡ ◇∃x (¬E!x & ◇E!x))›
4410 using "∀E"(1) "cqt:2[concrete]"[axiom_inst] by blast
4411 thus ?thesis using "thm-cont-e:1" "≡E" by blast
4412qed
4413
4414AOT_theorem "thm-cont-e:3": ‹◇∃x E!x›
4415proof (rule "CBF◇"[THEN "→E"])
4416 AOT_obtain a where ‹◇(E!a & ◇¬E!a)›
4417 using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF◇"[THEN "→E"]]] by blast
4418 AOT_hence ‹◇E!a›
4419 using "KBasic2:3"[THEN "→E", THEN "&E"(1)] by blast
4420 AOT_thus ‹∃x ◇E!x› using "∃I" by fast
4421qed
4422
4423AOT_theorem "thm-cont-e:4": ‹◇∃x ¬E!x›
4424proof (rule "CBF◇"[THEN "→E"])
4425 AOT_obtain a where ‹◇(E!a & ◇¬E!a)›
4426 using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF◇"[THEN "→E"]]] by blast
4427 AOT_hence ‹◇◇¬E!a›
4428 using "KBasic2:3"[THEN "→E", THEN "&E"(2)] by blast
4429 AOT_hence ‹◇¬E!a›
4430 using "4◇"[THEN "→E"] by blast
4431 AOT_thus ‹∃x ◇¬E!x› using "∃I" by fast
4432qed
4433
4434AOT_theorem "thm-cont-e:5": ‹Contingent([E!])›
4435proof -
4436 AOT_have ‹∀F (Contingent([F]) ≡ ◇∃x [F]x & ◇∃x ¬[F]x)›
4437 using "thm-cont-prop:2" GEN by fast
4438 AOT_hence ‹Contingent([E!]) ≡ ◇∃x E!x & ◇∃x ¬E!x›
4439 using "∀E"(1) "cqt:2[concrete]"[axiom_inst] by blast
4440 thus ?thesis
4441 using "thm-cont-e:3" "thm-cont-e:4" "≡E"(2) "&I" by blast
4442qed
4443
4444AOT_theorem "thm-cont-e:6": ‹Contingent([E!]⇧-)›
4445proof -
4446 AOT_have ‹∀F (Contingent([«F::<κ>»]) ≡ Contingent([F]⇧-))›
4447 using "thm-cont-prop:3" GEN by fast
4448 AOT_hence ‹Contingent([E!]) ≡ Contingent([E!]⇧-)›
4449 using "∀E" "cqt:2[concrete]"[axiom_inst] by fast
4450 thus ?thesis using "thm-cont-e:5" "≡E" by blast
4451qed
4452
4453AOT_theorem "thm-cont-e:7": ‹∃F∃G (Contingent([«F::<κ>»]) & Contingent([G]) & F ≠ G)›
4454proof (rule "∃I")+
4455 AOT_have ‹∀F [«F::<κ>»] ≠ [F]⇧-› using "thm-relation-negation:5" GEN by fast
4456 AOT_hence ‹[E!] ≠ [E!]⇧-›
4457 using "∀E" "cqt:2[concrete]"[axiom_inst] by fast
4458 AOT_thus ‹Contingent([E!]) & Contingent([E!]⇧-) & [E!] ≠ [E!]⇧-›
4459 using "thm-cont-e:5" "thm-cont-e:6" "&I" by metis
4460next
4461 AOT_show ‹E!⇧-↓›
4462 by (fact AOT)
4463next
4464 AOT_show ‹E!↓› by (fact "cqt:2[concrete]"[axiom_inst])
4465qed
4466
4467AOT_theorem "property-facts:1": ‹NonContingent([F]) → ¬∃G (Contingent([G]) & G = F)›
4468proof (rule "→I"; rule "raa-cor:2")
4469 AOT_assume ‹NonContingent([F])›
4470 AOT_hence 1: ‹Necessary([F]) ∨ Impossible([F])›
4471 using "contingent-properties:3"[THEN "≡⇩d⇩fE"] by blast
4472 AOT_assume ‹∃G (Contingent([G]) & G = F)›
4473 then AOT_obtain G where ‹Contingent([G]) & G = F› using "∃E"[rotated] by blast
4474 AOT_hence ‹Contingent([F])› using "rule=E" "&E" by blast
4475 AOT_hence ‹¬(Necessary([F]) ∨ Impossible([F]))›
4476 using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
4477 AOT_thus ‹(Necessary([F]) ∨ Impossible([F])) & ¬(Necessary([F]) ∨ Impossible([F]))›
4478 using 1 "&I" by blast
4479qed
4480
4481AOT_theorem "property-facts:2": ‹Contingent([F]) → ¬∃G (NonContingent([G]) & G = F)›
4482proof (rule "→I"; rule "raa-cor:2")
4483 AOT_assume ‹Contingent([F])›
4484 AOT_hence 1: ‹¬(Necessary([F]) ∨ Impossible([F]))›
4485 using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
4486 AOT_assume ‹∃G (NonContingent([G]) & G = F)›
4487 then AOT_obtain G where ‹NonContingent([G]) & G = F› using "∃E"[rotated] by blast
4488 AOT_hence ‹NonContingent([F])› using "rule=E" "&E" by blast
4489 AOT_hence ‹Necessary([F]) ∨ Impossible([F])›
4490 using "contingent-properties:3"[THEN "≡⇩d⇩fE"] by blast
4491 AOT_thus ‹(Necessary([F]) ∨ Impossible([F])) & ¬(Necessary([F]) ∨ Impossible([F]))›
4492 using 1 "&I" by blast
4493qed
4494
4495AOT_theorem "property-facts:3": ‹L ≠ [L]⇧- & L ≠ E! & L ≠ E!⇧- & [L]⇧- ≠ [E!]⇧- & E! ≠ [E!]⇧-›
4496proof -
4497 AOT_have noneqI: ‹Π ≠ Π'› if ‹φ{Π}› and ‹¬φ{Π'}› for φ and Π Π' :: ‹<κ>›
4498 apply (rule "=-infix"[THEN "≡⇩d⇩fI"]; rule "raa-cor:2")
4499 using "rule=E"[where φ=φ and τ=Π and σ = Π'] that "&I" by blast
4500 AOT_have contingent_denotes: ‹Π↓› if ‹Contingent([Π])› for Π :: ‹<κ>›
4501 using that "contingent-properties:4"[THEN "≡⇩d⇩fE", THEN "&E"(1)] by blast
4502 AOT_have not_noncontingent_if_contingent: ‹¬NonContingent([Π])› if ‹Contingent([Π])› for Π :: ‹<κ>›
4503 proof(rule RAA(2))
4504 AOT_show ‹¬(Necessary([Π]) ∨ Impossible([Π]))›
4505 using that "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1), OF contingent_denotes[OF that], THEN "≡E"(1)] by blast
4506 next
4507 AOT_assume ‹NonContingent([Π])›
4508 AOT_thus ‹Necessary([Π]) ∨ Impossible([Π])›
4509 using "contingent-properties:3"[THEN "≡⇩d⇩fE"] by blast
4510 qed
4511
4512 show ?thesis
4513 proof (safe intro!: "&I")
4514 AOT_show ‹L ≠ [L]⇧-›
4515 apply (rule "=⇩d⇩fI"(2)[OF L_def])
4516 apply "cqt:2[lambda]"
4517 apply (rule "∀E"(1)[where φ="λ Π . «Π ≠ [Π]⇧-»"])
4518 apply (rule GEN) apply (fact AOT)
4519 by "cqt:2[lambda]"
4520 next
4521 AOT_show ‹L ≠ E!›
4522 apply (rule noneqI)
4523 using "thm-noncont-e-e:3" not_noncontingent_if_contingent[OF "thm-cont-e:5"]
4524 by auto
4525 next
4526 AOT_show ‹L ≠ E!⇧-›
4527 apply (rule noneqI)
4528 using "thm-noncont-e-e:3" apply fast
4529 apply (rule not_noncontingent_if_contingent)
4530 apply (rule "∀E"(1)[where φ="λ Π . «Contingent([Π]) ≡ Contingent([Π]⇧-)»", rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
4531 using "thm-cont-prop:3" GEN apply fast
4532 using "thm-cont-e:5" by fast+
4533 next
4534 AOT_show ‹[L]⇧- ≠ E!⇧-›
4535 apply (rule noneqI)
4536 using "thm-noncont-e-e:4" apply fast
4537 apply (rule not_noncontingent_if_contingent)
4538 apply (rule "∀E"(1)[where φ="λ Π . «Contingent([Π]) ≡ Contingent([Π]⇧-)»", rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
4539 using "thm-cont-prop:3" GEN apply fast
4540 using "thm-cont-e:5" by fast+
4541 next
4542 AOT_show ‹E! ≠ E!⇧-›
4543 apply (rule "=⇩d⇩fI"(2)[OF L_def])
4544 apply "cqt:2[lambda]"
4545 apply (rule "∀E"(1)[where φ="λ Π . «Π ≠ [Π]⇧-»"])
4546 apply (rule GEN) apply (fact AOT)
4547 by (fact "cqt:2[concrete]"[axiom_inst])
4548 qed
4549qed
4550
4551AOT_theorem "thm-cont-propos:1": ‹NonContingent0(p) ≡ NonContingent0(((p)⇧-))›
4552proof(rule "≡I"; rule "→I")
4553 AOT_assume ‹NonContingent0(p)›
4554 AOT_hence ‹Necessary0(p) ∨ Impossible0(p)›
4555 using "contingent-properties:3[zero]"[THEN "≡⇩d⇩fE"] by blast
4556 moreover {
4557 AOT_assume ‹Necessary0(p)›
4558 AOT_hence 1: ‹□p› using "contingent-properties:1[zero]"[THEN "≡⇩d⇩fE"] by blast
4559 AOT_have ‹□¬((p)⇧-)›
4560 by (AOT_subst ‹¬((p)⇧-)› ‹p›)
4561 (auto simp add: 1 "thm-relation-negation:4")
4562 AOT_hence ‹Impossible0(((p)⇧-))›
4563 by (rule "contingent-properties:2[zero]"[THEN "≡⇩d⇩fI"])
4564 }
4565 moreover {
4566 AOT_assume ‹Impossible0(p)›
4567 AOT_hence 1: ‹□¬p›
4568 by (rule "contingent-properties:2[zero]"[THEN "≡⇩d⇩fE"])
4569 AOT_have ‹□((p)⇧-)›
4570 by (AOT_subst ‹((p)⇧-)› ‹¬p›)
4571 (auto simp: 1 "thm-relation-negation:3")
4572 AOT_hence ‹Necessary0(((p)⇧-))›
4573 by (rule "contingent-properties:1[zero]"[THEN "≡⇩d⇩fI"])
4574 }
4575 ultimately AOT_have ‹Necessary0(((p)⇧-)) ∨ Impossible0(((p)⇧-))›
4576 using "∨E"(1) "∨I" "→I" by metis
4577 AOT_thus ‹NonContingent0(((p)⇧-))›
4578 using "contingent-properties:3[zero]"[THEN "≡⇩d⇩fI"] by blast
4579next
4580 AOT_assume ‹NonContingent0(((p)⇧-))›
4581 AOT_hence ‹Necessary0(((p)⇧-)) ∨ Impossible0(((p)⇧-))›
4582 using "contingent-properties:3[zero]"[THEN "≡⇩d⇩fE"] by blast
4583 moreover {
4584 AOT_assume ‹Impossible0(((p)⇧-))›
4585 AOT_hence 1: ‹□¬((p)⇧-)›
4586 by (rule "contingent-properties:2[zero]"[THEN "≡⇩d⇩fE"])
4587 AOT_have ‹□p›
4588 by (AOT_subst (reverse) ‹p› ‹¬((p)⇧-)›)
4589 (auto simp: 1 "thm-relation-negation:4")
4590 AOT_hence ‹Necessary0(p)›
4591 using "contingent-properties:1[zero]"[THEN "≡⇩d⇩fI"] by blast
4592 }
4593 moreover {
4594 AOT_assume ‹Necessary0(((p)⇧-))›
4595 AOT_hence 1: ‹□((p)⇧-)›
4596 by (rule "contingent-properties:1[zero]"[THEN "≡⇩d⇩fE"])
4597 AOT_have ‹□¬p›
4598 by (AOT_subst (reverse) ‹¬p› ‹((p)⇧-)›)
4599 (auto simp: 1 "thm-relation-negation:3")
4600 AOT_hence ‹Impossible0(p)›
4601 by (rule "contingent-properties:2[zero]"[THEN "≡⇩d⇩fI"])
4602 }
4603 ultimately AOT_have ‹Necessary0(p) ∨ Impossible0(p)›
4604 using "∨E"(1) "∨I" "→I" by metis
4605 AOT_thus ‹NonContingent0(p)›
4606 using "contingent-properties:3[zero]"[THEN "≡⇩d⇩fI"] by blast
4607qed
4608
4609AOT_theorem "thm-cont-propos:2": ‹Contingent0(φ) ≡ ◇φ & ◇¬φ›
4610proof -
4611 AOT_have ‹Contingent0(φ) ≡ ¬(Necessary0(φ) ∨ Impossible0(φ))›
4612 using "contingent-properties:4[zero]"[THEN "≡Df"] by simp
4613 also AOT_have ‹… ≡ ¬Necessary0(φ) & ¬Impossible0(φ)›
4614 by (fact AOT)
4615 also AOT_have ‹… ≡ ¬Impossible0(φ) & ¬Necessary0(φ)›
4616 by (fact AOT)
4617 also AOT_have ‹… ≡ ◇φ & ◇¬φ›
4618 apply (AOT_subst ‹◇φ› ‹¬□¬φ›)
4619 apply (simp add: "conventions:5" "≡Df")
4620 apply (AOT_subst ‹Impossible0(φ)› ‹□¬φ›)
4621 apply (simp add: "contingent-properties:2[zero]" "≡Df")
4622 apply (AOT_subst (reverse) ‹◇¬φ› ‹¬□φ›)
4623 apply (simp add: "KBasic:11")
4624 apply (AOT_subst ‹Necessary0(φ)› ‹□φ›)
4625 apply (simp add: "contingent-properties:1[zero]" "≡Df")
4626 by (simp add: "oth-class-taut:3:a")
4627 finally show ?thesis.
4628qed
4629
4630AOT_theorem "thm-cont-propos:3": ‹Contingent0(p) ≡ Contingent0(((p)⇧-))›
4631proof -
4632 AOT_have ‹Contingent0(p) ≡ ◇p & ◇¬p› using "thm-cont-propos:2".
4633 also AOT_have ‹… ≡ ◇¬p & ◇p› by (fact AOT)
4634 also AOT_have ‹… ≡ ◇((p)⇧-) & ◇p›
4635 by (AOT_subst ‹((p)⇧-)› ‹¬p›)
4636 (auto simp: "thm-relation-negation:3" "oth-class-taut:3:a")
4637 also AOT_have ‹… ≡ ◇((p)⇧-) & ◇¬((p)⇧-)›
4638 by (AOT_subst ‹¬((p)⇧-)› ‹p›)
4639 (auto simp: "thm-relation-negation:4" "oth-class-taut:3:a")
4640 also AOT_have ‹… ≡ Contingent0(((p)⇧-))›
4641 using "thm-cont-propos:2"[symmetric] by blast
4642 finally show ?thesis.
4643qed
4644
4645AOT_define noncontingent_prop :: ‹φ› ("p⇩0")
4646 p⇩0_def: "(p⇩0) =⇩d⇩f (∀x (E!x → E!x))"
4647
4648AOT_theorem "thm-noncont-propos:1": ‹Necessary0((p⇩0))›
4649proof(rule "contingent-properties:1[zero]"[THEN "≡⇩d⇩fI"])
4650 AOT_show ‹□(p⇩0)›
4651 apply (rule "=⇩d⇩fI"(2)[OF p⇩0_def])
4652 using "log-prop-prop:2" apply simp
4653 using "if-p-then-p" RN GEN by fast
4654qed
4655
4656AOT_theorem "thm-noncont-propos:2": ‹Impossible0(((p⇩0)⇧-))›
4657proof(rule "contingent-properties:2[zero]"[THEN "≡⇩d⇩fI"])
4658 AOT_show ‹□¬((p⇩0)⇧-)›
4659 apply (AOT_subst ‹((p⇩0)⇧-)› ‹¬p⇩0›)
4660 using "thm-relation-negation:3" GEN "∀E"(1)[rotated, OF "log-prop-prop:2"] apply fast
4661 apply (AOT_subst (reverse) ‹¬¬p⇩0› ‹p⇩0›)
4662 apply (simp add: "oth-class-taut:3:b")
4663 apply (rule "=⇩d⇩fI"(2)[OF p⇩0_def])
4664 using "log-prop-prop:2" apply simp
4665 using "if-p-then-p" RN GEN by fast
4666qed
4667
4668AOT_theorem "thm-noncont-propos:3": ‹NonContingent0((p⇩0))›
4669 apply(rule "contingent-properties:3[zero]"[THEN "≡⇩d⇩fI"])
4670 using "thm-noncont-propos:1" "∨I" by blast
4671
4672AOT_theorem "thm-noncont-propos:4": ‹NonContingent0(((p⇩0)⇧-))›
4673 apply(rule "contingent-properties:3[zero]"[THEN "≡⇩d⇩fI"])
4674 using "thm-noncont-propos:2" "∨I" by blast
4675
4676AOT_theorem "thm-noncont-propos:5": ‹∃p∃q (NonContingent0((p)) & NonContingent0((q)) & p ≠ q)›
4677proof(rule "∃I")+
4678 AOT_have 0: ‹φ ≠ (φ)⇧-› for φ
4679 using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4680 AOT_thus ‹NonContingent0((p⇩0)) & NonContingent0(((p⇩0)⇧-)) & (p⇩0) ≠ (p⇩0)⇧-›
4681 using "thm-noncont-propos:3" "thm-noncont-propos:4" "&I" by auto
4682qed(auto simp: "log-prop-prop:2")
4683
4684AOT_act_theorem "no-cnac": ‹¬∃x(E!x & ¬❙𝒜E!x)›
4685proof(rule "raa-cor:2")
4686 AOT_assume ‹∃x(E!x & ¬❙𝒜E!x)›
4687 then AOT_obtain a where a: ‹E!a & ¬❙𝒜E!a›
4688 using "∃E"[rotated] by blast
4689 AOT_hence ‹❙𝒜¬E!a› using "&E" "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
4690 AOT_hence ‹¬E!a› using "logic-actual"[act_axiom_inst, THEN "→E"] by blast
4691 AOT_hence ‹E!a & ¬E!a› using a "&E" "&I" by blast
4692 AOT_thus ‹p & ¬p› for p using "raa-cor:1" by blast
4693qed
4694
4695AOT_theorem "pos-not-pna:1": ‹¬❙𝒜∃x (E!x & ¬❙𝒜E!x)›
4696proof(rule "raa-cor:2")
4697 AOT_assume ‹❙𝒜∃x (E!x & ¬❙𝒜E!x)›
4698 AOT_hence ‹∃x ❙𝒜(E!x & ¬❙𝒜E!x)›
4699 using "Act-Basic:10"[THEN "≡E"(1)] by blast
4700 then AOT_obtain a where ‹❙𝒜(E!a & ¬❙𝒜E!a)› using "∃E"[rotated] by blast
4701 AOT_hence 1: ‹❙𝒜E!a & ❙𝒜¬❙𝒜E!a› using "Act-Basic:2"[THEN "≡E"(1)] by blast
4702 AOT_hence ‹¬❙𝒜❙𝒜E!a› using "&E"(2) "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by blast
4703 AOT_hence ‹¬❙𝒜E!a› using "logic-actual-nec:4"[axiom_inst, THEN "≡E"(1)] RAA by blast
4704 AOT_thus ‹p & ¬p› for p using 1[THEN "&E"(1)] "&I" "raa-cor:1" by blast
4705qed
4706
4707AOT_theorem "pos-not-pna:2": ‹◇¬∃x(E!x & ¬❙𝒜E!x)›
4708proof (rule RAA(1))
4709 AOT_show ‹¬❙𝒜∃x (E!x & ¬❙𝒜E!x)› using "pos-not-pna:1" by blast
4710next
4711 AOT_assume ‹¬◇¬∃x (E!x & ¬❙𝒜E!x)›
4712 AOT_hence ‹□∃x (E!x & ¬❙𝒜E!x)›
4713 using "KBasic:12"[THEN "≡E"(2)] by blast
4714 AOT_thus ‹❙𝒜∃x (E!x & ¬❙𝒜E!x)›
4715 using "nec-imp-act"[THEN "→E"] by blast
4716qed
4717
4718AOT_theorem "pos-not-pna:3": ‹∃x (◇E!x & ¬❙𝒜E!x)›
4719proof -
4720 AOT_obtain a where ‹◇(E!a & ¬❙𝒜E!a)›
4721 using "qml:4"[axiom_inst] "BF◇"[THEN "→E"] "∃E"[rotated] by blast
4722 AOT_hence θ: ‹◇E!a› and ξ: ‹◇¬❙𝒜E!a› using "KBasic2:3"[THEN "→E"] "&E" by blast+
4723 AOT_have ‹¬□❙𝒜E!a› using ξ "KBasic:11"[THEN "≡E"(2)] by blast
4724 AOT_hence ‹¬❙𝒜E!a› using "Act-Basic:6"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)] by blast
4725 AOT_hence ‹◇E!a & ¬❙𝒜E!a› using θ "&I" by blast
4726 thus ?thesis using "∃I" by fast
4727qed
4728
4729AOT_define contingent_prop :: φ ("q⇩0")
4730 q⇩0_def: ‹(q⇩0) =⇩d⇩f (∃x (E!x & ¬❙𝒜E!x))›
4731
4732AOT_theorem q⇩0_prop: ‹◇q⇩0 & ◇¬q⇩0›
4733 apply (rule "=⇩d⇩fI"(2)[OF q⇩0_def])
4734 apply (fact "log-prop-prop:2")
4735 apply (rule "&I")
4736 apply (fact "qml:4"[axiom_inst])
4737 by (fact "pos-not-pna:2")
4738
4739AOT_theorem "basic-prop:1": ‹Contingent0((q⇩0))›
4740proof(rule "contingent-properties:4[zero]"[THEN "≡⇩d⇩fI"])
4741 AOT_have ‹¬Necessary0((q⇩0)) & ¬Impossible0((q⇩0))›
4742 proof (rule "&I"; rule "=⇩d⇩fI"(2)[OF q⇩0_def]; (rule "log-prop-prop:2" | rule "raa-cor:2"))
4743 AOT_assume ‹Necessary0(∃x (E!x & ¬❙𝒜E!x))›
4744 AOT_hence ‹□∃x (E!x & ¬❙𝒜E!x)›
4745 using "contingent-properties:1[zero]"[THEN "≡⇩d⇩fE"] by blast
4746 AOT_hence ‹❙𝒜∃x (E!x & ¬❙𝒜E!x)›
4747 using "Act-Basic:8"[THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
4748 AOT_thus ‹❙𝒜∃x (E!x & ¬❙𝒜E!x) & ¬❙𝒜∃x (E!x & ¬❙𝒜E!x)›
4749 using "pos-not-pna:1" "&I" by blast
4750 next
4751 AOT_assume ‹Impossible0(∃x (E!x & ¬❙𝒜E!x))›
4752 AOT_hence ‹□¬(∃x (E!x & ¬❙𝒜E!x))›
4753 using "contingent-properties:2[zero]"[THEN "≡⇩d⇩fE"] by blast
4754 AOT_hence ‹¬◇(∃x (E!x & ¬❙𝒜E!x))› using "KBasic2:1"[THEN "≡E"(1)] by blast
4755 AOT_thus ‹◇(∃x (E!x & ¬❙𝒜E!x)) & ¬◇(∃x (E!x & ¬❙𝒜E!x))›
4756 using "qml:4"[axiom_inst] "&I" by blast
4757 qed
4758 AOT_thus ‹¬(Necessary0((q⇩0)) ∨ Impossible0((q⇩0)))›
4759 using "oth-class-taut:5:d" "≡E"(2) by blast
4760qed
4761
4762AOT_theorem "basic-prop:2": ‹∃p Contingent0((p))›
4763 using "∃I"(1)[rotated, OF "log-prop-prop:2"] "basic-prop:1" by blast
4764
4765AOT_theorem "basic-prop:3": ‹Contingent0(((q⇩0)⇧-))›
4766 apply (AOT_subst ‹((q⇩0)⇧-)› ‹¬q⇩0›)
4767 apply (insert "thm-relation-negation:3" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"]; fast)
4768 apply (rule "contingent-properties:4[zero]"[THEN "≡⇩d⇩fI"])
4769 apply (rule "oth-class-taut:5:d"[THEN "≡E"(2)])
4770 apply (rule "&I")
4771 apply (rule "contingent-properties:1[zero]"[THEN "df-rules-formulas[3]", THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
4772 apply (rule "conventions:5"[THEN "≡⇩d⇩fE"])
4773 apply (rule "=⇩d⇩fE"(2)[OF q⇩0_def])
4774 apply (rule "log-prop-prop:2")
4775 apply (rule q⇩0_prop[THEN "&E"(1)])
4776 apply (rule "contingent-properties:2[zero]"[THEN "df-rules-formulas[3]", THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
4777 apply (rule "conventions:5"[THEN "≡⇩d⇩fE"])
4778 by (rule q⇩0_prop[THEN "&E"(2)])
4779
4780AOT_theorem "basic-prop:4": ‹∃p∃q (p ≠ q & Contingent0(p) & Contingent0(q))›
4781proof(rule "∃I")+
4782 AOT_have 0: ‹φ ≠ (φ)⇧-› for φ
4783 using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4784 AOT_show ‹(q⇩0) ≠ (q⇩0)⇧- & Contingent0(q⇩0) & Contingent0(((q⇩0)⇧-))›
4785 using "basic-prop:1" "basic-prop:3" "&I" 0 by presburger
4786qed(auto simp: "log-prop-prop:2")
4787
4788AOT_theorem "proposition-facts:1": ‹NonContingent0(p) → ¬∃q (Contingent0(q) & q = p)›
4789proof(rule "→I"; rule "raa-cor:2")
4790 AOT_assume ‹NonContingent0(p)›
4791 AOT_hence 1: ‹Necessary0(p) ∨ Impossible0(p)›
4792 using "contingent-properties:3[zero]"[THEN "≡⇩d⇩fE"] by blast
4793 AOT_assume ‹∃q (Contingent0(q) & q = p)›
4794 then AOT_obtain q where ‹Contingent0(q) & q = p› using "∃E"[rotated] by blast
4795 AOT_hence ‹Contingent0(p)› using "rule=E" "&E" by fast
4796 AOT_thus ‹(Necessary0(p) ∨ Impossible0(p)) & ¬(Necessary0(p) ∨ Impossible0(p))›
4797 using "contingent-properties:4[zero]"[THEN "≡⇩d⇩fE"] 1 "&I" by blast
4798qed
4799
4800AOT_theorem "proposition-facts:2": ‹Contingent0(p) → ¬∃q (NonContingent0(q) & q = p)›
4801proof(rule "→I"; rule "raa-cor:2")
4802 AOT_assume ‹Contingent0(p)›
4803 AOT_hence 1: ‹¬(Necessary0(p) ∨ Impossible0(p))›
4804 using "contingent-properties:4[zero]"[THEN "≡⇩d⇩fE"] by blast
4805 AOT_assume ‹∃q (NonContingent0(q) & q = p)›
4806 then AOT_obtain q where ‹NonContingent0(q) & q = p› using "∃E"[rotated] by blast
4807 AOT_hence ‹NonContingent0(p)› using "rule=E" "&E" by fast
4808 AOT_thus ‹(Necessary0(p) ∨ Impossible0(p)) & ¬(Necessary0(p) ∨ Impossible0(p))›
4809 using "contingent-properties:3[zero]"[THEN "≡⇩d⇩fE"] 1 "&I" by blast
4810qed
4811
4812AOT_theorem "proposition-facts:3": ‹(p⇩0) ≠ (p⇩0)⇧- & (p⇩0) ≠ (q⇩0) & (p⇩0) ≠ (q⇩0)⇧- & (p⇩0)⇧- ≠ (q⇩0)⇧- & (q⇩0) ≠ (q⇩0)⇧-›
4813proof -
4814 {
4815 fix χ φ ψ
4816 AOT_assume ‹χ{φ}›
4817 moreover AOT_assume ‹¬χ{ψ}›
4818 ultimately AOT_have ‹¬(χ{φ} ≡ χ{ψ})›
4819 using RAA "≡E" by metis
4820 moreover {
4821 AOT_have ‹∀p∀q ((¬(χ{p} ≡ χ{q})) → p ≠ q)›
4822 by (rule "∀I"; rule "∀I"; rule "pos-not-equiv-ne:4[zero]")
4823 AOT_hence ‹((¬(χ{φ} ≡ χ{ψ})) → φ ≠ ψ)›
4824 using "∀E" "log-prop-prop:2" by blast
4825 }
4826 ultimately AOT_have ‹φ ≠ ψ›
4827 using "→E" by blast
4828 } note 0 = this
4829 AOT_have contingent_neg: ‹Contingent0(φ) ≡ Contingent0(((φ)⇧-))› for φ
4830 using "thm-cont-propos:3" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4831 AOT_have not_noncontingent_if_contingent: ‹¬NonContingent0(φ)› if ‹Contingent0(φ)› for φ
4832 apply (rule "contingent-properties:3[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4833 using that "contingent-properties:4[zero]"[THEN "≡⇩d⇩fE"] by blast
4834 show ?thesis
4835 apply (rule "&I")+
4836 using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] apply fast
4837 apply (rule 0)
4838 using "thm-noncont-propos:3" apply fast
4839 apply (rule not_noncontingent_if_contingent)
4840 apply (fact AOT)
4841 apply (rule 0)
4842 apply (rule "thm-noncont-propos:3")
4843 apply (rule not_noncontingent_if_contingent)
4844 apply (rule contingent_neg[THEN "≡E"(1)])
4845 apply (fact AOT)
4846 apply (rule 0)
4847 apply (rule "thm-noncont-propos:4")
4848 apply (rule not_noncontingent_if_contingent)
4849 apply (rule contingent_neg[THEN "≡E"(1)])
4850 apply (fact AOT)
4851 using "thm-relation-negation:6" "∀I" "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
4852qed
4853
4854AOT_define ContingentlyTrue :: ‹φ ⇒ φ› ("ContingentlyTrue'(_')")
4855 "cont-tf:1": ‹ContingentlyTrue(p) ≡⇩d⇩f p & ◇¬p›
4856
4857AOT_define ContingentlyFalse :: ‹φ ⇒ φ› ("ContingentlyFalse'(_')")
4858 "cont-tf:2": ‹ContingentlyFalse(p) ≡⇩d⇩f ¬p & ◇p›
4859
4860AOT_theorem "cont-true-cont:1": ‹ContingentlyTrue((p)) → Contingent0((p))›
4861proof(rule "→I")
4862 AOT_assume ‹ContingentlyTrue((p))›
4863 AOT_hence 1: ‹p› and 2: ‹◇¬p› using "cont-tf:1"[THEN "≡⇩d⇩fE"] "&E" by blast+
4864 AOT_have ‹¬Necessary0((p))›
4865 apply (rule "contingent-properties:1[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4866 using 2 "KBasic:11"[THEN "≡E"(2)] by blast
4867 moreover AOT_have ‹¬Impossible0((p))›
4868 apply (rule "contingent-properties:2[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4869 apply (rule "conventions:5"[THEN "≡⇩d⇩fE"])
4870 using "T◇"[THEN "→E", OF 1].
4871 ultimately AOT_have ‹¬(Necessary0((p)) ∨ Impossible0((p)))›
4872 using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
4873 AOT_thus ‹Contingent0((p))›
4874 using "contingent-properties:4[zero]"[THEN "≡⇩d⇩fI"] by blast
4875qed
4876
4877AOT_theorem "cont-true-cont:2": ‹ContingentlyFalse((p)) → Contingent0((p))›
4878proof(rule "→I")
4879 AOT_assume ‹ContingentlyFalse((p))›
4880 AOT_hence 1: ‹¬p› and 2: ‹◇p› using "cont-tf:2"[THEN "≡⇩d⇩fE"] "&E" by blast+
4881 AOT_have ‹¬Necessary0((p))›
4882 apply (rule "contingent-properties:1[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4883 using "KBasic:11"[THEN "≡E"(2)] "T◇"[THEN "→E", OF 1] by blast
4884 moreover AOT_have ‹¬Impossible0((p))›
4885 apply (rule "contingent-properties:2[zero]"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
4886 apply (rule "conventions:5"[THEN "≡⇩d⇩fE"])
4887 using 2.
4888 ultimately AOT_have ‹¬(Necessary0((p)) ∨ Impossible0((p)))›
4889 using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
4890 AOT_thus ‹Contingent0((p))›
4891 using "contingent-properties:4[zero]"[THEN "≡⇩d⇩fI"] by blast
4892qed
4893
4894AOT_theorem "cont-true-cont:3": ‹ContingentlyTrue((p)) ≡ ContingentlyFalse(((p)⇧-))›
4895proof(rule "≡I"; rule "→I")
4896 AOT_assume ‹ContingentlyTrue((p))›
4897 AOT_hence 0: ‹p & ◇¬p› using "cont-tf:1"[THEN "≡⇩d⇩fE"] by blast
4898 AOT_have 1: ‹ContingentlyFalse(¬p)›
4899 apply (rule "cont-tf:2"[THEN "≡⇩d⇩fI"])
4900 apply (AOT_subst (reverse) ‹¬¬p› p)
4901 by (auto simp: "oth-class-taut:3:b" 0)
4902 AOT_show ‹ContingentlyFalse(((p)⇧-))›
4903 apply (AOT_subst ‹((p)⇧-)› ‹¬p›)
4904 by (auto simp: "thm-relation-negation:3" 1)
4905next
4906 AOT_assume 1: ‹ContingentlyFalse(((p)⇧-))›
4907 AOT_have ‹ContingentlyFalse(¬p)›
4908 by (AOT_subst (reverse) ‹¬p› ‹((p)⇧-)›)
4909 (auto simp: "thm-relation-negation:3" 1)
4910 AOT_hence ‹¬¬p & ◇¬p› using "cont-tf:2"[THEN "≡⇩d⇩fE"] by blast
4911 AOT_hence ‹p & ◇¬p›
4912 using "&I" "&E" "useful-tautologies:1"[THEN "→E"] by metis
4913 AOT_thus ‹ContingentlyTrue((p))›
4914 using "cont-tf:1"[THEN "≡⇩d⇩fI"] by blast
4915qed
4916
4917AOT_theorem "cont-true-cont:4": ‹ContingentlyFalse((p)) ≡ ContingentlyTrue(((p)⇧-))›
4918proof(rule "≡I"; rule "→I")
4919 AOT_assume ‹ContingentlyFalse(p)›
4920 AOT_hence 0: ‹¬p & ◇p›
4921 using "cont-tf:2"[THEN "≡⇩d⇩fE"] by blast
4922 AOT_have ‹¬p & ◇¬¬p›
4923 by (AOT_subst (reverse) ‹¬¬p› p)
4924 (auto simp: "oth-class-taut:3:b" 0)
4925 AOT_hence 1: ‹ContingentlyTrue(¬p)›
4926 by (rule "cont-tf:1"[THEN "≡⇩d⇩fI"])
4927 AOT_show ‹ContingentlyTrue(((p)⇧-))›
4928 by (AOT_subst ‹((p)⇧-)› ‹¬p›)
4929 (auto simp: "thm-relation-negation:3" 1)
4930next
4931 AOT_assume 1: ‹ContingentlyTrue(((p)⇧-))›
4932 AOT_have ‹ContingentlyTrue(¬p)›
4933 by (AOT_subst (reverse) ‹¬p› ‹((p)⇧-)›)
4934 (auto simp add: "thm-relation-negation:3" 1)
4935 AOT_hence 2: ‹¬p & ◇¬¬p› using "cont-tf:1"[THEN "≡⇩d⇩fE"] by blast
4936 AOT_have ‹◇p›
4937 by (AOT_subst p ‹¬¬p›)
4938 (auto simp add: "oth-class-taut:3:b" 2[THEN "&E"(2)])
4939 AOT_hence ‹¬p & ◇p› using 2[THEN "&E"(1)] "&I" by blast
4940 AOT_thus ‹ContingentlyFalse(p)›
4941 by (rule "cont-tf:2"[THEN "≡⇩d⇩fI"])
4942qed
4943
4944AOT_theorem "cont-true-cont:5": ‹(ContingentlyTrue((p)) & Necessary0((q))) → p ≠ q›
4945proof (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
4946 AOT_assume ‹ContingentlyTrue((p))›
4947 AOT_hence ‹◇¬p›
4948 using "cont-tf:1"[THEN "≡⇩d⇩fE"] "&E" by blast
4949 AOT_hence 0: ‹¬□p› using "KBasic:11"[THEN "≡E"(2)] by blast
4950 AOT_assume ‹Necessary0((q))›
4951 moreover AOT_assume ‹¬(p ≠ q)›
4952 AOT_hence ‹p = q›
4953 using "=-infix"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4954 "useful-tautologies:1"[THEN "→E"] by blast
4955 ultimately AOT_have ‹Necessary0((p))› using "rule=E" id_sym by blast
4956 AOT_hence ‹□p›
4957 using "contingent-properties:1[zero]"[THEN "≡⇩d⇩fE"] by blast
4958 AOT_thus ‹□p & ¬□p› using 0 "&I" by blast
4959qed
4960
4961AOT_theorem "cont-true-cont:6": ‹(ContingentlyFalse((p)) & Impossible0((q))) → p ≠ q›
4962proof (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
4963 AOT_assume ‹ContingentlyFalse((p))›
4964 AOT_hence ‹◇p›
4965 using "cont-tf:2"[THEN "≡⇩d⇩fE"] "&E" by blast
4966 AOT_hence 1: ‹¬□¬p›
4967 using "conventions:5"[THEN "≡⇩d⇩fE"] by blast
4968 AOT_assume ‹Impossible0((q))›
4969 moreover AOT_assume ‹¬(p ≠ q)›
4970 AOT_hence ‹p = q›
4971 using "=-infix"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
4972 "useful-tautologies:1"[THEN "→E"] by blast
4973 ultimately AOT_have ‹Impossible0((p))› using "rule=E" id_sym by blast
4974 AOT_hence ‹□¬p›
4975 using "contingent-properties:2[zero]"[THEN "≡⇩d⇩fE"] by blast
4976 AOT_thus ‹□¬p & ¬□¬p› using 1 "&I" by blast
4977qed
4978
4979AOT_act_theorem "q0cf:1": ‹ContingentlyFalse(q⇩0)›
4980 apply (rule "cont-tf:2"[THEN "≡⇩d⇩fI"])
4981 apply (rule "=⇩d⇩fI"(2)[OF q⇩0_def])
4982 apply (fact "log-prop-prop:2")
4983 apply (rule "&I")
4984 apply (fact "no-cnac")
4985 by (fact "qml:4"[axiom_inst])
4986
4987AOT_act_theorem "q0cf:2": ‹ContingentlyTrue(((q⇩0)⇧-))›
4988 apply (rule "cont-tf:1"[THEN "≡⇩d⇩fI"])
4989 apply (rule "=⇩d⇩fI"(2)[OF q⇩0_def])
4990 apply (fact "log-prop-prop:2")
4991 apply (rule "&I")
4992 apply (rule "thm-relation-negation:3"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(2)])
4993 apply (fact "no-cnac")
4994 apply (rule "rule=E"[rotated, OF "thm-relation-negation:7"[unvarify p, OF "log-prop-prop:2", THEN id_sym]])
4995 apply (AOT_subst (reverse) ‹¬¬(∃x (E!x & ¬❙𝒜E!x))› ‹∃x (E!x & ¬❙𝒜E!x)›)
4996 by (auto simp: "oth-class-taut:3:b" "qml:4"[axiom_inst])
4997
4998
4999
5000AOT_theorem "cont-tf-thm:1": ‹∃p ContingentlyTrue((p))›
5001proof(rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
5002 AOT_assume ‹q⇩0›
5003 AOT_hence ‹q⇩0 & ◇¬q⇩0› using q⇩0_prop[THEN "&E"(2)] "&I" by blast
5004 AOT_thus ‹ContingentlyTrue(q⇩0)›
5005 by (rule "cont-tf:1"[THEN "≡⇩d⇩fI"])
5006next
5007 AOT_assume ‹¬q⇩0›
5008 AOT_hence ‹¬q⇩0 & ◇q⇩0› using q⇩0_prop[THEN "&E"(1)] "&I" by blast
5009 AOT_hence ‹ContingentlyFalse(q⇩0)›
5010 by (rule "cont-tf:2"[THEN "≡⇩d⇩fI"])
5011 AOT_thus ‹ContingentlyTrue(((q⇩0)⇧-))›
5012 by (rule "cont-true-cont:4"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(1)])
5013qed(auto simp: "log-prop-prop:2")
5014
5015
5016AOT_theorem "cont-tf-thm:2": ‹∃p ContingentlyFalse((p))›
5017proof(rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
5018 AOT_assume ‹q⇩0›
5019 AOT_hence ‹q⇩0 & ◇¬q⇩0› using q⇩0_prop[THEN "&E"(2)] "&I" by blast
5020 AOT_hence ‹ContingentlyTrue(q⇩0)›
5021 by (rule "cont-tf:1"[THEN "≡⇩d⇩fI"])
5022 AOT_thus ‹ContingentlyFalse(((q⇩0)⇧-))›
5023 by (rule "cont-true-cont:3"[unvarify p, OF "log-prop-prop:2", THEN "≡E"(1)])
5024next
5025 AOT_assume ‹¬q⇩0›
5026 AOT_hence ‹¬q⇩0 & ◇q⇩0› using q⇩0_prop[THEN "&E"(1)] "&I" by blast
5027 AOT_thus ‹ContingentlyFalse(q⇩0)›
5028 by (rule "cont-tf:2"[THEN "≡⇩d⇩fI"])
5029qed(auto simp: "log-prop-prop:2")
5030
5031
5032AOT_theorem "property-facts1:1": ‹∃F∃x ([F]x & ◇¬[F]x)›
5033proof -
5034 fix x
5035 AOT_obtain p⇩1 where ‹ContingentlyTrue((p⇩1))›
5036 using "cont-tf-thm:1" "∃E"[rotated] by blast
5037 AOT_hence 1: ‹p⇩1 & ◇¬p⇩1› using "cont-tf:1"[THEN "≡⇩d⇩fE"] by blast
5038 AOT_modally_strict {
5039 AOT_have ‹for arbitrary p: ❙⊢⇩□ ([λz p]x ≡ p)›
5040 by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
5041 AOT_hence ‹for arbitrary p: ❙⊢⇩□ □ ([λz p]x ≡ p)›
5042 by (rule RN)
5043 AOT_hence ‹∀p □([λz p]x ≡ p)› using GEN by fast
5044 AOT_hence ‹□([λz p⇩1]x ≡ p⇩1)› using "∀E" by fast
5045 } note 2 = this
5046 AOT_hence ‹□([λz p⇩1]x ≡ p⇩1)› using "∀E" by blast
5047 AOT_hence ‹[λz p⇩1]x› using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
5048 moreover AOT_have ‹◇¬[λz p⇩1]x›
5049 apply (AOT_subst_using subst: 2[THEN "qml:2"[axiom_inst, THEN "→E"]])
5050 using 1[THEN "&E"(2)] by blast
5051 ultimately AOT_have ‹[λz p⇩1]x & ◇¬[λz p⇩1]x› using "&I" by blast
5052 AOT_hence ‹∃x ([λz p⇩1]x & ◇¬[λz p⇩1]x)› using "∃I"(2) by fast
5053 moreover AOT_have ‹[λz p⇩1]↓› by "cqt:2[lambda]"
5054 ultimately AOT_show ‹∃F∃x ([F]x & ◇¬[F]x)› by (rule "∃I"(1))
5055qed
5056
5057
5058AOT_theorem "property-facts1:2": ‹∃F∃x (¬[F]x & ◇[F]x)›
5059proof -
5060 fix x
5061 AOT_obtain p⇩1 where ‹ContingentlyFalse((p⇩1))›
5062 using "cont-tf-thm:2" "∃E"[rotated] by blast
5063 AOT_hence 1: ‹¬p⇩1 & ◇p⇩1› using "cont-tf:2"[THEN "≡⇩d⇩fE"] by blast
5064 AOT_modally_strict {
5065 AOT_have ‹for arbitrary p: ❙⊢⇩□ ([λz p]x ≡ p)›
5066 by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
5067 AOT_hence ‹for arbitrary p: ❙⊢⇩□ (¬[λz p]x ≡ ¬p)›
5068 using "oth-class-taut:4:b" "≡E" by blast
5069 AOT_hence ‹for arbitrary p: ❙⊢⇩□ □(¬[λz p]x ≡ ¬p)›
5070 by (rule RN)
5071 AOT_hence ‹∀p □(¬[λz p]x ≡ ¬p)› using GEN by fast
5072 AOT_hence ‹□(¬[λz p⇩1]x ≡ ¬p⇩1)› using "∀E" by fast
5073 } note 2 = this
5074 AOT_hence ‹□(¬[λz p⇩1]x ≡ ¬p⇩1)› using "∀E" by blast
5075 AOT_hence 3: ‹¬[λz p⇩1]x› using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
5076 AOT_modally_strict {
5077 AOT_have ‹for arbitrary p: ❙⊢⇩□ ([λz p]x ≡ p)›
5078 by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
5079 AOT_hence ‹for arbitrary p: ❙⊢⇩□ □([λz p]x ≡ p)›
5080 by (rule RN)
5081 AOT_hence ‹∀p □([λz p]x ≡ p)› using GEN by fast
5082 AOT_hence ‹□([λz p⇩1]x ≡ p⇩1)› using "∀E" by fast
5083 } note 4 = this
5084 AOT_have ‹◇[λz p⇩1]x›
5085 apply (AOT_subst_using subst: 4[THEN "qml:2"[axiom_inst, THEN "→E"]])
5086 using 1[THEN "&E"(2)] by blast
5087 AOT_hence ‹¬[λz p⇩1]x & ◇[λz p⇩1]x› using 3 "&I" by blast
5088 AOT_hence ‹∃x (¬[λz p⇩1]x & ◇[λz p⇩1]x)› using "∃I"(2) by fast
5089 moreover AOT_have ‹[λz p⇩1]↓› by "cqt:2[lambda]"
5090 ultimately AOT_show ‹∃F∃x (¬[F]x & ◇[F]x)› by (rule "∃I"(1))
5091qed
5092
5093context
5094begin
5095
5096private AOT_lemma eqnotnec_123_Aux_ζ: ‹[L]x ≡ (E!x → E!x)›
5097 apply (rule "=⇩d⇩fI"(2)[OF L_def])
5098 apply "cqt:2[lambda]"
5099 apply (rule "beta-C-meta"[THEN "→E"])
5100 by "cqt:2[lambda]"
5101
5102private AOT_lemma eqnotnec_123_Aux_ω: ‹[λz φ]x ≡ φ›
5103 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5104
5105private AOT_lemma eqnotnec_123_Aux_θ: ‹φ ≡ ∀x([L]x ≡ [λz φ]x)›
5106proof(rule "≡I"; rule "→I"; (rule "∀I")?)
5107 fix x
5108 AOT_assume 1: ‹φ›
5109 AOT_have ‹[L]x ≡ (E!x → E!x)› using eqnotnec_123_Aux_ζ.
5110 also AOT_have ‹… ≡ φ›
5111 using "if-p-then-p" 1 "≡I" "→I" by simp
5112 also AOT_have ‹… ≡ [λz φ]x›
5113 using "Commutativity of ≡"[THEN "≡E"(1)] eqnotnec_123_Aux_ω by blast
5114 finally AOT_show ‹[L]x ≡ [λz φ]x›.
5115next
5116 fix x
5117 AOT_assume ‹∀x([L]x ≡ [λz φ]x)›
5118 AOT_hence ‹[L]x ≡ [λz φ]x› using "∀E" by blast
5119 also AOT_have ‹… ≡ φ› using eqnotnec_123_Aux_ω.
5120 finally AOT_have ‹φ ≡ [L]x› using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5121 also AOT_have ‹… ≡ E!x → E!x› using eqnotnec_123_Aux_ζ.
5122 finally AOT_show ‹φ› using "≡E" "if-p-then-p" by fast
5123qed
5124private lemmas eqnotnec_123_Aux_ξ = eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
5125 THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
5126 THEN "RM◇"]
5127private lemmas eqnotnec_123_Aux_ξ' = eqnotnec_123_Aux_θ[THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)], THEN "RM◇"]
5128
5129AOT_theorem "eqnotnec:1": ‹∃F∃G(∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5130proof-
5131 AOT_obtain p⇩1 where ‹ContingentlyTrue(p⇩1)› using "cont-tf-thm:1" "∃E"[rotated] by blast
5132 AOT_hence ‹p⇩1 & ◇¬p⇩1› using "cont-tf:1"[THEN "≡⇩d⇩fE"] by blast
5133 AOT_hence ‹∀x ([L]x ≡ [λz p⇩1]x) & ◇¬∀x([L]x ≡ [λz p⇩1]x)›
5134 apply - apply (rule "&I")
5135 using "&E" eqnotnec_123_Aux_θ[THEN "≡E"(1)] eqnotnec_123_Aux_ξ "→E" by fast+
5136 AOT_hence ‹∃G (∀x([L]x ≡ [G]x) & ◇¬∀x([L]x ≡ [G]x))›
5137 by (rule "∃I") "cqt:2[lambda]"
5138 AOT_thus ‹∃F∃G (∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5139 apply (rule "∃I")
5140 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
5141qed
5142
5143AOT_theorem "eqnotnec:2": ‹∃F∃G(¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5144proof-
5145 AOT_obtain p⇩1 where ‹ContingentlyFalse(p⇩1)› using "cont-tf-thm:2" "∃E"[rotated] by blast
5146 AOT_hence ‹¬p⇩1 & ◇p⇩1› using "cont-tf:2"[THEN "≡⇩d⇩fE"] by blast
5147 AOT_hence ‹¬∀x ([L]x ≡ [λz p⇩1]x) & ◇∀x([L]x ≡ [λz p⇩1]x)›
5148 apply - apply (rule "&I")
5149 using "&E" eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)] eqnotnec_123_Aux_ξ' "→E" by fast+
5150 AOT_hence ‹∃G (¬∀x([L]x ≡ [G]x) & ◇∀x([L]x ≡ [G]x))›
5151 by (rule "∃I") "cqt:2[lambda]"
5152 AOT_thus ‹∃F∃G (¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5153 apply (rule "∃I")
5154 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
5155qed
5156
5157AOT_theorem "eqnotnec:3": ‹∃F∃G(❙𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5158proof-
5159 AOT_have ‹¬❙𝒜q⇩0›
5160 apply (rule "=⇩d⇩fI"(2)[OF q⇩0_def])
5161 apply (fact "log-prop-prop:2")
5162 by (fact AOT)
5163 AOT_hence ‹❙𝒜¬q⇩0›
5164 using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5165 AOT_hence ‹❙𝒜¬∀x ([L]x ≡ [λz q⇩0]x)›
5166 using eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
5167 THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
5168 THEN "RA[2]", THEN "act-cond"[THEN "→E"], THEN "→E"] by blast
5169 moreover AOT_have ‹◇∀x ([L]x ≡ [λz q⇩0]x)› using eqnotnec_123_Aux_ξ'[THEN "→E"] q⇩0_prop[THEN "&E"(1)] by blast
5170 ultimately AOT_have ‹❙𝒜¬∀x ([L]x ≡ [λz q⇩0]x) & ◇∀x ([L]x ≡ [λz q⇩0]x)› using "&I" by blast
5171 AOT_hence ‹∃G (❙𝒜¬∀x([L]x ≡ [G]x) & ◇∀x([L]x ≡ [G]x))›
5172 by (rule "∃I") "cqt:2[lambda]"
5173 AOT_thus ‹∃F∃G (❙𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5174 apply (rule "∃I")
5175 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
5176qed
5177
5178end
5179
5180
5182AOT_theorem "eqnotnec:4": ‹∀F∃G(∀x([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5183proof(rule GEN)
5184 fix F
5185
5186 AOT_have Aux_A: ‹❙⊢⇩□ ψ → ∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5187 proof(rule "→I"; rule GEN)
5188 AOT_modally_strict {
5189 fix x
5190 AOT_assume 0: ‹ψ›
5191 AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5192 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5193 also AOT_have ‹... ≡ [F]x›
5194 apply (rule "≡I"; rule "→I")
5195 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5196 using 0 "&I" by blast
5197 finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5198 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5199 }
5200 qed
5201
5202 AOT_have Aux_B: ‹❙⊢⇩□ ψ → ∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5203 proof (rule "→I"; rule GEN)
5204 AOT_modally_strict {
5205 fix x
5206 AOT_assume 0: ‹ψ›
5207 AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5208 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5209 also AOT_have ‹... ≡ [F]x›
5210 apply (rule "≡I"; rule "→I")
5211 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5212 apply (rule "∨I"(1)) using 0 "&I" by blast
5213 finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5214 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5215 }
5216 qed
5217
5218 AOT_have Aux_C: ‹❙⊢⇩□ ◇¬ψ → ◇¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5219 proof(rule "RM◇"; rule "→I"; rule "raa-cor:2")
5220 AOT_modally_strict {
5221 AOT_assume 0: ‹¬ψ›
5222 AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5223 AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5224 moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5225 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5226 moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5227 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5228 ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5229 using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5230 moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5231 ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5232 AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5233 }
5234 qed
5235
5236 AOT_have Aux_D: ‹□∀z ([F]z ≡ [λz [F]z & ψ]z) → (◇¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ◇¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5237 proof (rule "→I")
5238 AOT_assume A: ‹□∀z([F]z ≡ [λz [F]z & ψ]z)›
5239 AOT_show ‹◇¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ◇¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5240 proof(rule "≡I"; rule "KBasic:13"[THEN "→E"];
5241 rule "RN[prem]"[where Γ="{«∀z([F]z ≡ [λz [F]z & ψ]z)»}", simplified];
5242 (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5243 AOT_modally_strict {
5244 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5245 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5246 AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5247 AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5248 AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5249 AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5250 }
5251 next
5252 AOT_modally_strict {
5253 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5254 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5255 AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5256 AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5257 AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5258 AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5259 }
5260 qed(auto simp: A)
5261 qed
5262
5263 AOT_obtain p⇩1 where p⇩1_prop: ‹p⇩1 & ◇¬p⇩1› using "cont-tf-thm:1" "∃E"[rotated] "cont-tf:1"[THEN "≡⇩d⇩fE"] by blast
5264 {
5265 AOT_assume 1: ‹□∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5266 AOT_have 2: ‹∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)›
5267 using Aux_B[THEN "→E", OF p⇩1_prop[THEN "&E"(1)]].
5268 AOT_have ‹◇¬∀x([λz [F]z & p⇩1]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)›
5269 using Aux_C[THEN "→E", OF p⇩1_prop[THEN "&E"(2)]].
5270 AOT_hence 3: ‹◇¬∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)›
5271 using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5272 AOT_hence ‹∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x) & ◇¬∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)› using 2 "&I" by blast
5273 AOT_hence ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5274 by (rule "∃I"(1)) "cqt:2[lambda]"
5275 }
5276 moreover {
5277 AOT_assume 2: ‹¬□∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5278 AOT_hence ‹◇¬∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5279 using "KBasic:11"[THEN "≡E"(1)] by blast
5280 AOT_hence ‹∀x ([F]x ≡ [λz [F]z & p⇩1]x) & ◇¬∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5281 using Aux_A[THEN "→E", OF p⇩1_prop[THEN "&E"(1)]] "&I" by blast
5282 AOT_hence ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5283 by (rule "∃I"(1)) "cqt:2[lambda]"
5284 }
5285 ultimately AOT_show ‹∃G (∀x ([F]x ≡ [G]x) & ◇¬∀x([F]x ≡ [G]x))›
5286 using "∨E"(1)[OF "exc-mid"] "→I" by blast
5287qed
5288
5289AOT_theorem "eqnotnec:5": ‹∀F∃G(¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5290proof(rule GEN)
5291 fix F
5292
5293 AOT_have Aux_A: ‹❙⊢⇩□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5294 proof(rule "RM◇"; rule "→I"; rule GEN)
5295 AOT_modally_strict {
5296 fix x
5297 AOT_assume 0: ‹ψ›
5298 AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5299 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5300 also AOT_have ‹... ≡ [F]x›
5301 apply (rule "≡I"; rule "→I")
5302 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5303 using 0 "&I" by blast
5304 finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5305 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5306 }
5307 qed
5308
5309 AOT_have Aux_B: ‹❙⊢⇩□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5310 proof (rule "RM◇"; rule "→I"; rule GEN)
5311 AOT_modally_strict {
5312 fix x
5313 AOT_assume 0: ‹ψ›
5314 AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5315 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5316 also AOT_have ‹... ≡ [F]x›
5317 apply (rule "≡I"; rule "→I")
5318 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5319 apply (rule "∨I"(1)) using 0 "&I" by blast
5320 finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5321 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5322 }
5323 qed
5324
5325 AOT_have Aux_C: ‹❙⊢⇩□ ¬ψ → ¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5326 proof(rule "→I"; rule "raa-cor:2")
5327 AOT_modally_strict {
5328 AOT_assume 0: ‹¬ψ›
5329 AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5330 AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5331 moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5332 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5333 moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5334 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5335 ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5336 using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5337 moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5338 ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5339 AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5340 }
5341 qed
5342
5343 AOT_have Aux_D: ‹∀z ([F]z ≡ [λz [F]z & ψ]z) → (¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5344 proof (rule "→I"; rule "≡I"; (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5345 AOT_modally_strict {
5346 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5347 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5348 AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5349 AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5350 AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5351 AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5352 }
5353 next
5354 AOT_modally_strict {
5355 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5356 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5357 AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5358 AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5359 AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5360 AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5361 }
5362 qed
5363
5364 AOT_obtain p⇩1 where p⇩1_prop: ‹¬p⇩1 & ◇p⇩1› using "cont-tf-thm:2" "∃E"[rotated] "cont-tf:2"[THEN "≡⇩d⇩fE"] by blast
5365 {
5366 AOT_assume 1: ‹∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5367 AOT_have 2: ‹◇∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)›
5368 using Aux_B[THEN "→E", OF p⇩1_prop[THEN "&E"(2)]].
5369 AOT_have ‹¬∀x([λz [F]z & p⇩1]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)›
5370 using Aux_C[THEN "→E", OF p⇩1_prop[THEN "&E"(1)]].
5371 AOT_hence 3: ‹¬∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)›
5372 using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5373 AOT_hence ‹¬∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x) & ◇∀x([F]x ≡ [λz [F]z & p⇩1 ∨ ¬p⇩1]x)› using 2 "&I" by blast
5374 AOT_hence ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5375 by (rule "∃I"(1)) "cqt:2[lambda]"
5376 }
5377 moreover {
5378 AOT_assume 2: ‹¬∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5379 AOT_hence ‹¬∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5380 using "KBasic:11"[THEN "≡E"(1)] by blast
5381 AOT_hence ‹¬∀x ([F]x ≡ [λz [F]z & p⇩1]x) & ◇∀x([F]x ≡ [λz [F]z & p⇩1]x)›
5382 using Aux_A[THEN "→E", OF p⇩1_prop[THEN "&E"(2)]] "&I" by blast
5383 AOT_hence ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5384 by (rule "∃I"(1)) "cqt:2[lambda]"
5385 }
5386 ultimately AOT_show ‹∃G (¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5387 using "∨E"(1)[OF "exc-mid"] "→I" by blast
5388qed
5389
5390AOT_theorem "eqnotnec:6": ‹∀F∃G(❙𝒜¬∀x([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5391proof(rule GEN)
5392 fix F
5393
5394 AOT_have Aux_A: ‹❙⊢⇩□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
5395 proof(rule "RM◇"; rule "→I"; rule GEN)
5396 AOT_modally_strict {
5397 fix x
5398 AOT_assume 0: ‹ψ›
5399 AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
5400 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5401 also AOT_have ‹... ≡ [F]x›
5402 apply (rule "≡I"; rule "→I")
5403 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5404 using 0 "&I" by blast
5405 finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
5406 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5407 }
5408 qed
5409
5410 AOT_have Aux_B: ‹❙⊢⇩□ ◇ψ → ◇∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
5411 proof (rule "RM◇"; rule "→I"; rule GEN)
5412 AOT_modally_strict {
5413 fix x
5414 AOT_assume 0: ‹ψ›
5415 AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
5416 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5417 also AOT_have ‹... ≡ [F]x›
5418 apply (rule "≡I"; rule "→I")
5419 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E" apply blast
5420 apply (rule "∨I"(1)) using 0 "&I" by blast
5421 finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
5422 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
5423 }
5424 qed
5425
5426 AOT_have Aux_C: ‹❙⊢⇩□ ❙𝒜¬ψ → ❙𝒜¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
5427 proof(rule "act-cond"[THEN "→E"]; rule "RA[2]"; rule "→I"; rule "raa-cor:2")
5428 AOT_modally_strict {
5429 AOT_assume 0: ‹¬ψ›
5430 AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
5431 AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5432 moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
5433 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5434 moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5435 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5436 ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
5437 using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
5438 moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
5439 ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
5440 AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
5441 }
5442 qed
5443
5444 AOT_have Aux_D: ‹❙𝒜∀z ([F]z ≡ [λz [F]z & ψ]z) → (❙𝒜¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ❙𝒜¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
5445 proof (rule "→I"; rule "Act-Basic:5"[THEN "≡E"(1)])
5446 AOT_assume ‹❙𝒜∀z ([F]z ≡ [λz [F]z & ψ]z)›
5447 AOT_thus ‹❙𝒜(¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))›
5448 proof (rule "RA[3]"[where Γ="{«∀z ([F]z ≡ [λz [F]z & ψ]z)»}", simplified, rotated])
5449 AOT_modally_strict {
5450 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5451 AOT_thus ‹¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡ ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5452 apply -
5453 proof(rule "≡I"; (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
5454 AOT_modally_strict {
5455 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5456 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5457 AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5458 AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5459 AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "≡E" 1 2 by meson
5460 AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5461 }
5462 next
5463 AOT_modally_strict {
5464 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
5465 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z using "∀E" by blast
5466 AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
5467 AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using "∀E" by blast
5468 AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z using 1 2 "≡E" by meson
5469 AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› by (rule GEN)
5470 }
5471 qed
5472 }
5473 qed
5474 qed
5475
5476 AOT_have ‹¬❙𝒜q⇩0›
5477 apply (rule "=⇩d⇩fI"(2)[OF q⇩0_def])
5478 apply (fact "log-prop-prop:2")
5479 by (fact AOT)
5480 AOT_hence q⇩0_prop_1: ‹❙𝒜¬q⇩0›
5481 using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5482 {
5483 AOT_assume 1: ‹❙𝒜∀x([F]x ≡ [λz [F]z & q⇩0]x)›
5484 AOT_have 2: ‹◇∀x([F]x ≡ [λz [F]z & q⇩0 ∨ ¬q⇩0]x)›
5485 using Aux_B[THEN "→E", OF q⇩0_prop[THEN "&E"(1)]].
5486 AOT_have ‹❙𝒜¬∀x([λz [F]z & q⇩0]x ≡ [λz [F]z & q⇩0 ∨ ¬q⇩0]x)›
5487 using Aux_C[THEN "→E", OF q⇩0_prop_1].
5488 AOT_hence 3: ‹❙𝒜¬∀x([F]x ≡ [λz [F]z & q⇩0 ∨ ¬q⇩0]x)›
5489 using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
5490 AOT_hence ‹❙𝒜¬∀x([F]x ≡ [λz [F]z & q⇩0 ∨ ¬q⇩0]x) & ◇∀x([F]x ≡ [λz [F]z & q⇩0 ∨ ¬q⇩0]x)› using 2 "&I" by blast
5491 AOT_hence ‹∃G (❙𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5492 by (rule "∃I"(1)) "cqt:2[lambda]"
5493 }
5494 moreover {
5495 AOT_assume 2: ‹¬❙𝒜∀x([F]x ≡ [λz [F]z & q⇩0]x)›
5496 AOT_hence ‹❙𝒜¬∀x([F]x ≡ [λz [F]z & q⇩0]x)›
5497 using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
5498 AOT_hence ‹❙𝒜¬∀x ([F]x ≡ [λz [F]z & q⇩0]x) & ◇∀x([F]x ≡ [λz [F]z & q⇩0]x)›
5499 using Aux_A[THEN "→E", OF q⇩0_prop[THEN "&E"(1)]] "&I" by blast
5500 AOT_hence ‹∃G (❙𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5501 by (rule "∃I"(1)) "cqt:2[lambda]"
5502 }
5503 ultimately AOT_show ‹∃G (❙𝒜¬∀x ([F]x ≡ [G]x) & ◇∀x([F]x ≡ [G]x))›
5504 using "∨E"(1)[OF "exc-mid"] "→I" by blast
5505qed
5506
5507AOT_theorem "oa-contingent:1": ‹O! ≠ A!›
5508proof(rule "≡⇩d⇩fI"[OF "=-infix"]; rule "raa-cor:2")
5509 fix x
5510 AOT_assume 1: ‹O! = A!›
5511 AOT_hence ‹[λx ◇E!x] = A!›
5512 by (rule "=⇩d⇩fE"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5513 AOT_hence ‹[λx ◇E!x] = [λx ¬◇E!x]›
5514 by (rule "=⇩d⇩fE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5515 moreover AOT_have ‹[λx ◇E!x]x ≡ ◇E!x›
5516 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5517 ultimately AOT_have ‹[λx ¬◇E!x]x ≡ ◇E!x›
5518 using "rule=E" by fast
5519 moreover AOT_have ‹[λx ¬◇E!x]x ≡ ¬◇E!x›
5520 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5521 ultimately AOT_have ‹◇E!x ≡ ¬◇E!x› using "≡E"(6) "Commutativity of ≡"[THEN "≡E"(1)] by blast
5522 AOT_thus "(◇E!x ≡ ¬◇E!x) & ¬(◇E!x ≡ ¬◇E!x)" using "oth-class-taut:3:c" "&I" by blast
5523qed
5524
5525AOT_theorem "oa-contingent:2": ‹O!x ≡ ¬A!x›
5526proof -
5527 AOT_have ‹O!x ≡ [λx ◇E!x]x›
5528 apply (rule "≡I"; rule "→I")
5529 apply (rule "=⇩d⇩fE"(2)[OF AOT_ordinary])
5530 apply "cqt:2[lambda]"
5531 apply argo
5532 apply (rule "=⇩d⇩fI"(2)[OF AOT_ordinary])
5533 apply "cqt:2[lambda]"
5534 by argo
5535 also AOT_have ‹… ≡ ◇E!x›
5536 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
5537 also AOT_have ‹… ≡ ¬¬◇E!x›
5538 using "oth-class-taut:3:b".
5539 also AOT_have ‹… ≡ ¬[λx ¬◇E!x]x›
5540 by (rule "beta-C-meta"[THEN "→E", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric]) "cqt:2[lambda]"
5541 also AOT_have ‹… ≡ ¬A!x›
5542 apply (rule "≡I"; rule "→I")
5543 apply (rule "=⇩d⇩fI"(2)[OF AOT_abstract])
5544 apply "cqt:2[lambda]"
5545 apply argo
5546 apply (rule "=⇩d⇩fE"(2)[OF AOT_abstract])
5547 apply "cqt:2[lambda]"
5548 by argo
5549 finally show ?thesis.
5550qed
5551
5552AOT_theorem "oa-contingent:3": ‹A!x ≡ ¬O!x›
5553 by (AOT_subst ‹A!x› ‹¬¬A!x›)
5554 (auto simp add: "oth-class-taut:3:b" "oa-contingent:2"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric])
5555
5556AOT_theorem "oa-contingent:4": ‹Contingent(O!)›
5557proof (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:1", THEN "≡E"(2)]; rule "&I")
5558 AOT_have ‹◇∃x E!x› using "thm-cont-e:3" .
5559 AOT_hence ‹∃x ◇E!x› using "BF◇"[THEN "→E"] by blast
5560 then AOT_obtain a where ‹◇E!a› using "∃E"[rotated] by blast
5561 AOT_hence ‹[λx ◇E!x]a›
5562 by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
5563 AOT_hence ‹O!a›
5564 by (rule "=⇩d⇩fI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5565 AOT_hence ‹∃x O!x› using "∃I" by blast
5566 AOT_thus ‹◇∃x O!x› using "T◇"[THEN "→E"] by blast
5567next
5568 AOT_obtain a where ‹A!a›
5569 using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5570 AOT_hence ‹¬O!a› using "oa-contingent:3"[THEN "≡E"(1)] by blast
5571 AOT_hence ‹∃x ¬O!x› using "∃I" by fast
5572 AOT_thus ‹◇∃x ¬O!x› using "T◇"[THEN "→E"] by blast
5573qed
5574
5575AOT_theorem "oa-contingent:5": ‹Contingent(A!)›
5576proof (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:2", THEN "≡E"(2)]; rule "&I")
5577 AOT_obtain a where ‹A!a›
5578 using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5579 AOT_hence ‹∃x A!x› using "∃I" by fast
5580 AOT_thus ‹◇∃x A!x› using "T◇"[THEN "→E"] by blast
5581next
5582 AOT_have ‹◇∃x E!x› using "thm-cont-e:3" .
5583 AOT_hence ‹∃x ◇E!x› using "BF◇"[THEN "→E"] by blast
5584 then AOT_obtain a where ‹◇E!a› using "∃E"[rotated] by blast
5585 AOT_hence ‹[λx ◇E!x]a›
5586 by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
5587 AOT_hence ‹O!a›
5588 by (rule "=⇩d⇩fI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
5589 AOT_hence ‹¬A!a› using "oa-contingent:2"[THEN "≡E"(1)] by blast
5590 AOT_hence ‹∃x ¬A!x› using "∃I" by fast
5591 AOT_thus ‹◇∃x ¬A!x› using "T◇"[THEN "→E"] by blast
5592qed
5593
5594AOT_theorem "oa-contingent:7": ‹O!⇧-x ≡ ¬A!⇧-x›
5595proof -
5596 AOT_have ‹O!x ≡ ¬A!x›
5597 using "oa-contingent:2" by blast
5598 also AOT_have ‹… ≡ A!⇧-x›
5599 using "thm-relation-negation:1"[symmetric, unvarify F, OF "oa-exist:2"].
5600 finally AOT_have 1: ‹O!x ≡ A!⇧-x›.
5601
5602 AOT_have ‹A!x ≡ ¬O!x›
5603 using "oa-contingent:3" by blast
5604 also AOT_have ‹… ≡ O!⇧-x›
5605 using "thm-relation-negation:1"[symmetric, unvarify F, OF "oa-exist:1"].
5606 finally AOT_have 2: ‹A!x ≡ O!⇧-x›.
5607
5608 AOT_show ‹O!⇧-x ≡ ¬A!⇧-x›
5609 using 1[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]] "oa-contingent:3"[of _ x] 2[symmetric]
5610 "≡E"(5) by blast
5611qed
5612
5613AOT_theorem "oa-contingent:6": ‹O!⇧- ≠ A!⇧-›
5614proof (rule "=-infix"[THEN "≡⇩d⇩fI"]; rule "raa-cor:2")
5615 AOT_assume 1: ‹O!⇧- = A!⇧-›
5616 fix x
5617 AOT_have ‹A!⇧-x ≡ O!⇧-x›
5618 apply (rule "rule=E"[rotated, OF 1]) by (fact "oth-class-taut:3:a")
5619 AOT_hence ‹A!⇧-x ≡ ¬A!⇧-x›
5620 using "oa-contingent:7" "≡E" by fast
5621 AOT_thus ‹(A!⇧-x ≡ ¬A!⇧-x) & ¬(A!⇧-x ≡ ¬A!⇧-x)› using "oth-class-taut:3:c" "&I" by blast
5622qed
5623
5624AOT_theorem "oa-contingent:8": ‹Contingent(O!⇧-)›
5625 using "thm-cont-prop:3"[unvarify F, OF "oa-exist:1", THEN "≡E"(1), OF "oa-contingent:4"].
5626
5627AOT_theorem "oa-contingent:9": ‹Contingent(A!⇧-)›
5628 using "thm-cont-prop:3"[unvarify F, OF "oa-exist:2", THEN "≡E"(1), OF "oa-contingent:5"].
5629
5630AOT_define WeaklyContingent :: ‹Π ⇒ φ› ("WeaklyContingent'(_')")
5631 "df-cont-nec": "WeaklyContingent([F]) ≡⇩d⇩f Contingent([F]) & ∀x (◇[F]x → □[F]x)"
5632
5633AOT_theorem "cont-nec-fact1:1": ‹WeaklyContingent([F]) ≡ WeaklyContingent([F]⇧-)›
5634proof -
5635 AOT_have ‹WeaklyContingent([F]) ≡ Contingent([F]) & ∀x (◇[F]x → □[F]x)›
5636 using "df-cont-nec"[THEN "≡Df"] by blast
5637 also AOT_have ‹... ≡ Contingent([F]⇧-) & ∀x (◇[F]x → □[F]x)›
5638 apply (rule "oth-class-taut:8:f"[THEN "≡E"(2)]; rule "→I")
5639 using "thm-cont-prop:3".
5640 also AOT_have ‹… ≡ Contingent([F]⇧-) & ∀x (◇[F]⇧-x → □[F]⇧-x)›
5641 proof (rule "oth-class-taut:8:e"[THEN "≡E"(2)]; rule "→I"; rule "≡I"; rule "→I"; rule GEN; rule "→I")
5642 fix x
5643 AOT_assume 0: ‹∀x (◇[F]x → □[F]x)›
5644 AOT_assume 1: ‹◇[F]⇧-x›
5645 AOT_have ‹◇¬[F]x›
5646 by (AOT_subst (reverse) ‹¬[F]x› ‹[F]⇧-x›)
5647 (auto simp add: "thm-relation-negation:1" 1)
5648 AOT_hence 2: ‹¬□[F]x›
5649 using "KBasic:11"[THEN "≡E"(2)] by blast
5650 AOT_show ‹□[F]⇧-x›
5651 proof (rule "raa-cor:1")
5652 AOT_assume 3: ‹¬□[F]⇧-x›
5653 AOT_have ‹¬□¬[F]x›
5654 by (AOT_subst (reverse) ‹¬[F]x› ‹[F]⇧-x›)
5655 (auto simp add: "thm-relation-negation:1" 3)
5656 AOT_hence ‹◇[F]x›
5657 using "conventions:5"[THEN "≡⇩d⇩fI"] by simp
5658 AOT_hence ‹□[F]x› using 0 "∀E" "→E" by fast
5659 AOT_thus ‹□[F]x & ¬□[F]x› using "&I" 2 by blast
5660 qed
5661 next
5662 fix x
5663 AOT_assume 0: ‹∀x (◇[F]⇧-x → □[F]⇧-x)›
5664 AOT_assume 1: ‹◇[F]x›
5665 AOT_have ‹◇¬[F]⇧-x›
5666 by (AOT_subst ‹¬[F]⇧-x› ‹[F]x›)
5667 (auto simp: "thm-relation-negation:2" 1)
5668 AOT_hence 2: ‹¬□[F]⇧-x›
5669 using "KBasic:11"[THEN "≡E"(2)] by blast
5670 AOT_show ‹□[F]x›
5671 proof (rule "raa-cor:1")
5672 AOT_assume 3: ‹¬□[F]x›
5673 AOT_have ‹¬□¬[F]⇧-x›
5674 by (AOT_subst ‹¬[F]⇧-x› ‹[F]x›)
5675 (auto simp add: "thm-relation-negation:2" 3)
5676 AOT_hence ‹◇[F]⇧-x›
5677 using "conventions:5"[THEN "≡⇩d⇩fI"] by simp
5678 AOT_hence ‹□[F]⇧-x› using 0 "∀E" "→E" by fast
5679 AOT_thus ‹□[F]⇧-x & ¬□[F]⇧-x› using "&I" 2 by blast
5680 qed
5681 qed
5682 also AOT_have ‹… ≡ WeaklyContingent([F]⇧-)›
5683 using "df-cont-nec"[THEN "≡Df", symmetric] by blast
5684 finally show ?thesis.
5685qed
5686
5687AOT_theorem "cont-nec-fact1:2": ‹(WeaklyContingent([F]) & ¬WeaklyContingent([G])) → F ≠ G›
5688proof (rule "→I"; rule "=-infix"[THEN "≡⇩d⇩fI"]; rule "raa-cor:2")
5689 AOT_assume 1: ‹WeaklyContingent([F]) & ¬WeaklyContingent([G])›
5690 AOT_hence ‹WeaklyContingent([F])› using "&E" by blast
5691 moreover AOT_assume ‹F = G›
5692 ultimately AOT_have ‹WeaklyContingent([G])›
5693 using "rule=E" by blast
5694 AOT_thus ‹WeaklyContingent([G]) & ¬WeaklyContingent([G])›
5695 using 1 "&I" "&E" by blast
5696qed
5697
5698AOT_theorem "cont-nec-fact2:1": ‹WeaklyContingent(O!)›
5699proof (rule "df-cont-nec"[THEN "≡⇩d⇩fI"]; rule "&I")
5700 AOT_show ‹Contingent(O!)›
5701 using "oa-contingent:4".
5702next
5703 AOT_show ‹∀x (◇[O!]x → □[O!]x)›
5704 apply (rule GEN; rule "→I")
5705 using "oa-facts:5"[THEN "≡E"(1)] by blast
5706qed
5707
5708
5709AOT_theorem "cont-nec-fact2:2": ‹WeaklyContingent(A!)›
5710proof (rule "df-cont-nec"[THEN "≡⇩d⇩fI"]; rule "&I")
5711 AOT_show ‹Contingent(A!)›
5712 using "oa-contingent:5".
5713next
5714 AOT_show ‹∀x (◇[A!]x → □[A!]x)›
5715 apply (rule GEN; rule "→I")
5716 using "oa-facts:6"[THEN "≡E"(1)] by blast
5717qed
5718
5719AOT_theorem "cont-nec-fact2:3": ‹¬WeaklyContingent(E!)›
5720proof (rule "df-cont-nec"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)];
5721 rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "raa-cor:2")
5722 AOT_have ‹◇∃x (E!x & ¬❙𝒜E!x)› using "qml:4"[axiom_inst].
5723 AOT_hence ‹∃x ◇(E!x & ¬❙𝒜E!x)› using "BF◇"[THEN "→E"] by blast
5724 then AOT_obtain a where ‹◇(E!a & ¬❙𝒜E!a)› using "∃E"[rotated] by blast
5725 AOT_hence 1: ‹◇E!a & ◇¬❙𝒜E!a› using "KBasic2:3"[THEN "→E"] by simp
5726 moreover AOT_assume ‹∀x (◇[E!]x → □[E!]x)›
5727 ultimately AOT_have ‹□E!a› using "&E" "∀E" "→E" by fast
5728 AOT_hence ‹❙𝒜E!a› using "nec-imp-act"[THEN "→E"] by blast
5729 AOT_hence ‹□❙𝒜E!a› using "qml-act:1"[axiom_inst, THEN "→E"] by blast
5730 moreover AOT_have ‹¬□❙𝒜E!a› using "KBasic:11"[THEN "≡E"(2)] 1[THEN "&E"(2)] by meson
5731 ultimately AOT_have ‹□❙𝒜E!a & ¬□❙𝒜E!a› using "&I" by blast
5732 AOT_thus ‹p & ¬p› for p using "raa-cor:1" by blast
5733qed
5734
5735AOT_theorem "cont-nec-fact2:4": ‹¬WeaklyContingent(L)›
5736 apply (rule "df-cont-nec"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)];
5737 rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(1))
5738 apply (rule "contingent-properties:4"[THEN "≡Df", THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(2)])
5739 apply (rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "useful-tautologies:2"[THEN "→E"])
5740 using "thm-noncont-e-e:3"[THEN "contingent-properties:3"[THEN "≡⇩d⇩fE"]].
5741
5742
5743AOT_theorem "cont-nec-fact2:5": ‹O! ≠ E! & O! ≠ E!⇧- & O! ≠ L & O! ≠ L⇧-›
5744proof -
5745 AOT_have 1: ‹L↓›
5746 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
5747 {
5748 fix φ and Π Π' :: ‹<κ>›
5749 AOT_have A: ‹¬(φ{Π'} ≡ φ{Π})› if ‹φ{Π}› and ‹¬φ{Π'}›
5750 proof (rule "raa-cor:2")
5751 AOT_assume ‹φ{Π'} ≡ φ{Π}›
5752 AOT_hence ‹φ{Π'}› using that(1) "≡E" by blast
5753 AOT_thus ‹φ{Π'} & ¬φ{Π'}› using that(2) "&I" by blast
5754 qed
5755 AOT_have ‹Π' ≠ Π› if ‹Π↓› and ‹Π'↓› and ‹φ{Π}› and ‹¬φ{Π'}›
5756 using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E", OF that(1,2), OF A[OF that(3, 4)]].
5757 } note 0 = this
5758 show ?thesis
5759 apply(safe intro!: "&I"; rule 0)
5760 using "cqt:2[concrete]"[axiom_inst] apply blast
5761 using "oa-exist:1" apply blast
5762 using "cont-nec-fact2:3" apply fast
5763 apply (rule "useful-tautologies:2"[THEN "→E"])
5764 using "cont-nec-fact2:1" apply fast
5765 using "rel-neg-T:3" apply fast
5766 using "oa-exist:1" apply blast
5767 using "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:3", OF "cqt:2[concrete]"[axiom_inst]] apply fast
5768 apply (rule "useful-tautologies:2"[THEN "→E"])
5769 using "cont-nec-fact2:1" apply blast
5770 apply (rule "=⇩d⇩fI"(2)[OF L_def]; "cqt:2[lambda]")
5771 using "oa-exist:1" apply fast
5772 using "cont-nec-fact2:4" apply fast
5773 apply (rule "useful-tautologies:2"[THEN "→E"])
5774 using "cont-nec-fact2:1" apply fast
5775 using "rel-neg-T:3" apply fast
5776 using "oa-exist:1" apply fast
5777 apply (rule "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
5778 apply (rule "=⇩d⇩fI"(2)[OF L_def]; "cqt:2[lambda]")
5779 apply (rule "useful-tautologies:2"[THEN "→E"])
5780 using "cont-nec-fact2:1" by blast
5781qed
5782
5783
5784AOT_theorem "cont-nec-fact2:6": ‹A! ≠ E! & A! ≠ E!⇧- & A! ≠ L & A! ≠ L⇧-›
5785proof -
5786 AOT_have 1: ‹L↓›
5787 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
5788 {
5789 fix φ and Π Π' :: ‹<κ>›
5790 AOT_have A: ‹¬(φ{Π'} ≡ φ{Π})› if ‹φ{Π}› and ‹¬φ{Π'}›
5791 proof (rule "raa-cor:2")
5792 AOT_assume ‹φ{Π'} ≡ φ{Π}›
5793 AOT_hence ‹φ{Π'}› using that(1) "≡E" by blast
5794 AOT_thus ‹φ{Π'} & ¬φ{Π'}› using that(2) "&I" by blast
5795 qed
5796 AOT_have ‹Π' ≠ Π› if ‹Π↓› and ‹Π'↓› and ‹φ{Π}› and ‹¬φ{Π'}›
5797 using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E", OF that(1,2), OF A[OF that(3, 4)]].
5798 } note 0 = this
5799 show ?thesis
5800 apply(safe intro!: "&I"; rule 0)
5801 using "cqt:2[concrete]"[axiom_inst] apply blast
5802 using "oa-exist:2" apply blast
5803 using "cont-nec-fact2:3" apply fast
5804 apply (rule "useful-tautologies:2"[THEN "→E"])
5805 using "cont-nec-fact2:2" apply fast
5806 using "rel-neg-T:3" apply fast
5807 using "oa-exist:2" apply blast
5808 using "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:3", OF "cqt:2[concrete]"[axiom_inst]] apply fast
5809 apply (rule "useful-tautologies:2"[THEN "→E"])
5810 using "cont-nec-fact2:2" apply blast
5811 apply (rule "=⇩d⇩fI"(2)[OF L_def]; "cqt:2[lambda]")
5812 using "oa-exist:2" apply fast
5813 using "cont-nec-fact2:4" apply fast
5814 apply (rule "useful-tautologies:2"[THEN "→E"])
5815 using "cont-nec-fact2:2" apply fast
5816 using "rel-neg-T:3" apply fast
5817 using "oa-exist:2" apply fast
5818 apply (rule "cont-nec-fact1:1"[unvarify F, THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
5819 apply (rule "=⇩d⇩fI"(2)[OF L_def]; "cqt:2[lambda]")
5820 apply (rule "useful-tautologies:2"[THEN "→E"])
5821 using "cont-nec-fact2:2" by blast
5822qed
5823
5824AOT_define necessary_or_contingently_false :: ‹φ ⇒ φ› ("❙Δ_" [49] 54)
5825 ‹❙Δp ≡⇩d⇩f □p ∨ (¬❙𝒜p & ◇p)›
5826
5827AOT_theorem sixteen:
5828 shows ‹∃F⇩1∃F⇩2∃F⇩3∃F⇩4∃F⇩5∃F⇩6∃F⇩7∃F⇩8∃F⇩9∃F⇩1⇩0∃F⇩1⇩1∃F⇩1⇩2∃F⇩1⇩3∃F⇩1⇩4∃F⇩1⇩5∃F⇩1⇩6 (
5829«F⇩1::<κ>» ≠ F⇩2 & F⇩1 ≠ F⇩3 & F⇩1 ≠ F⇩4 & F⇩1 ≠ F⇩5 & F⇩1 ≠ F⇩6 & F⇩1 ≠ F⇩7 & F⇩1 ≠ F⇩8 & F⇩1 ≠ F⇩9 & F⇩1 ≠ F⇩1⇩0 & F⇩1 ≠ F⇩1⇩1 & F⇩1 ≠ F⇩1⇩2 & F⇩1 ≠ F⇩1⇩3 & F⇩1 ≠ F⇩1⇩4 & F⇩1 ≠ F⇩1⇩5 & F⇩1 ≠ F⇩1⇩6 &
5830F⇩2 ≠ F⇩3 & F⇩2 ≠ F⇩4 & F⇩2 ≠ F⇩5 & F⇩2 ≠ F⇩6 & F⇩2 ≠ F⇩7 & F⇩2 ≠ F⇩8 & F⇩2 ≠ F⇩9 & F⇩2 ≠ F⇩1⇩0 & F⇩2 ≠ F⇩1⇩1 & F⇩2 ≠ F⇩1⇩2 & F⇩2 ≠ F⇩1⇩3 & F⇩2 ≠ F⇩1⇩4 & F⇩2 ≠ F⇩1⇩5 & F⇩2 ≠ F⇩1⇩6 &
5831F⇩3 ≠ F⇩4 & F⇩3 ≠ F⇩5 & F⇩3 ≠ F⇩6 & F⇩3 ≠ F⇩7 & F⇩3 ≠ F⇩8 & F⇩3 ≠ F⇩9 & F⇩3 ≠ F⇩1⇩0 & F⇩3 ≠ F⇩1⇩1 & F⇩3 ≠ F⇩1⇩2 & F⇩3 ≠ F⇩1⇩3 & F⇩3 ≠ F⇩1⇩4 & F⇩3 ≠ F⇩1⇩5 & F⇩3 ≠ F⇩1⇩6 &
5832F⇩4 ≠ F⇩5 & F⇩4 ≠ F⇩6 & F⇩4 ≠ F⇩7 & F⇩4 ≠ F⇩8 & F⇩4 ≠ F⇩9 & F⇩4 ≠ F⇩1⇩0 & F⇩4 ≠ F⇩1⇩1 & F⇩4 ≠ F⇩1⇩2 & F⇩4 ≠ F⇩1⇩3 & F⇩4 ≠ F⇩1⇩4 & F⇩4 ≠ F⇩1⇩5 & F⇩4 ≠ F⇩1⇩6 &
5833F⇩5 ≠ F⇩6 & F⇩5 ≠ F⇩7 & F⇩5 ≠ F⇩8 & F⇩5 ≠ F⇩9 & F⇩5 ≠ F⇩1⇩0 & F⇩5 ≠ F⇩1⇩1 & F⇩5 ≠ F⇩1⇩2 & F⇩5 ≠ F⇩1⇩3 & F⇩5 ≠ F⇩1⇩4 & F⇩5 ≠ F⇩1⇩5 & F⇩5 ≠ F⇩1⇩6 &
5834F⇩6 ≠ F⇩7 & F⇩6 ≠ F⇩8 & F⇩6 ≠ F⇩9 & F⇩6 ≠ F⇩1⇩0 & F⇩6 ≠ F⇩1⇩1 & F⇩6 ≠ F⇩1⇩2 & F⇩6 ≠ F⇩1⇩3 & F⇩6 ≠ F⇩1⇩4 & F⇩6 ≠ F⇩1⇩5 & F⇩6 ≠ F⇩1⇩6 &
5835F⇩7 ≠ F⇩8 & F⇩7 ≠ F⇩9 & F⇩7 ≠ F⇩1⇩0 & F⇩7 ≠ F⇩1⇩1 & F⇩7 ≠ F⇩1⇩2 & F⇩7 ≠ F⇩1⇩3 & F⇩7 ≠ F⇩1⇩4 & F⇩7 ≠ F⇩1⇩5 & F⇩7 ≠ F⇩1⇩6 &
5836F⇩8 ≠ F⇩9 & F⇩8 ≠ F⇩1⇩0 & F⇩8 ≠ F⇩1⇩1 & F⇩8 ≠ F⇩1⇩2 & F⇩8 ≠ F⇩1⇩3 & F⇩8 ≠ F⇩1⇩4 & F⇩8 ≠ F⇩1⇩5 & F⇩8 ≠ F⇩1⇩6 &
5837F⇩9 ≠ F⇩1⇩0 & F⇩9 ≠ F⇩1⇩1 & F⇩9 ≠ F⇩1⇩2 & F⇩9 ≠ F⇩1⇩3 & F⇩9 ≠ F⇩1⇩4 & F⇩9 ≠ F⇩1⇩5 & F⇩9 ≠ F⇩1⇩6 &
5838F⇩1⇩0 ≠ F⇩1⇩1 & F⇩1⇩0 ≠ F⇩1⇩2 & F⇩1⇩0 ≠ F⇩1⇩3 & F⇩1⇩0 ≠ F⇩1⇩4 & F⇩1⇩0 ≠ F⇩1⇩5 & F⇩1⇩0 ≠ F⇩1⇩6 &
5839F⇩1⇩1 ≠ F⇩1⇩2 & F⇩1⇩1 ≠ F⇩1⇩3 & F⇩1⇩1 ≠ F⇩1⇩4 & F⇩1⇩1 ≠ F⇩1⇩5 & F⇩1⇩1 ≠ F⇩1⇩6 &
5840F⇩1⇩2 ≠ F⇩1⇩3 & F⇩1⇩2 ≠ F⇩1⇩4 & F⇩1⇩2 ≠ F⇩1⇩5 & F⇩1⇩2 ≠ F⇩1⇩6 &
5841F⇩1⇩3 ≠ F⇩1⇩4 & F⇩1⇩3 ≠ F⇩1⇩5 & F⇩1⇩3 ≠ F⇩1⇩6 &
5842F⇩1⇩4 ≠ F⇩1⇩5 & F⇩1⇩4 ≠ F⇩1⇩6 &
5843F⇩1⇩5 ≠ F⇩1⇩6)›
5844proof -
5845
5846 AOT_have Delta_pos: ‹❙Δφ → ◇φ› for φ
5847 proof(rule "→I")
5848 AOT_assume ‹❙Δφ›
5849 AOT_hence ‹□φ ∨ (¬❙𝒜φ & ◇φ)›
5850 using "≡⇩d⇩fE"[OF necessary_or_contingently_false] by blast
5851 moreover {
5852 AOT_assume ‹□φ›
5853 AOT_hence ‹◇φ›
5854 by (metis "B◇" "T◇" "vdash-properties:10")
5855 }
5856 moreover {
5857 AOT_assume ‹¬❙𝒜φ & ◇φ›
5858 AOT_hence ‹◇φ›
5859 using "&E" by blast
5860 }
5861 ultimately AOT_show ‹◇φ›
5862 by (metis "∨E"(2) "raa-cor:1")
5863 qed
5864
5865 AOT_have act_and_not_nec_not_delta: ‹¬❙Δφ› if ‹❙𝒜φ› and ‹¬□φ› for φ
5866 using "≡⇩d⇩fE" "&E"(1) "∨E"(2) necessary_or_contingently_false "raa-cor:3" that(1) that(2) by blast
5867 AOT_have act_and_pos_not_not_delta: ‹¬❙Δφ› if ‹❙𝒜φ› and ‹◇¬φ› for φ
5868 using "KBasic:11" act_and_not_nec_not_delta "≡E"(2) that(1) that(2) by blast
5869 AOT_have impossible_delta: ‹¬❙Δφ› if ‹¬◇φ› for φ
5870 using Delta_pos "modus-tollens:1" that by blast
5871 AOT_have not_act_and_pos_delta: ‹❙Δφ› if ‹¬❙𝒜φ› and ‹◇φ› for φ
5872 by (meson "≡⇩d⇩fI" "&I" "∨I"(2) necessary_or_contingently_false that(1) that(2))
5873 AOT_have nec_delta: ‹❙Δφ› if ‹□φ› for φ
5874 using "≡⇩d⇩fI" "∨I"(1) necessary_or_contingently_false that by blast
5875
5876 AOT_obtain a where a_prop: ‹A!a›
5877 using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
5878 AOT_obtain b where b_prop: ‹◇[E!]b & ¬❙𝒜[E!]b›
5879 using "pos-not-pna:3" using "∃E"[rotated] by blast
5880
5881 AOT_have b_ord: ‹[O!]b›
5882 proof(rule "=⇩d⇩fI"(2)[OF AOT_ordinary])
5883 AOT_show ‹[λx ◇[E!]x]↓› by "cqt:2[lambda]"
5884 next
5885 AOT_show ‹[λx ◇[E!]x]b›
5886 proof (rule "β←C"(1); ("cqt:2[lambda]")?)
5887 AOT_show ‹b↓› by (rule "cqt:2[const_var]"[axiom_inst])
5888 AOT_show ‹◇[E!]b› by (fact b_prop[THEN "&E"(1)])
5889 qed
5890 qed
5891
5892 AOT_have nec_not_L_neg: ‹□¬[L⇧-]x› for x
5893 using "thm-noncont-e-e:2" "contingent-properties:2"[THEN "≡⇩d⇩fE"] "&E"
5894 CBF[THEN "→E"] "∀E" by blast
5895 AOT_have nec_L: ‹□[L]x› for x
5896 using "thm-noncont-e-e:1" "contingent-properties:1"[THEN "≡⇩d⇩fE"]
5897 CBF[THEN "→E"] "∀E" by blast
5898
5899 AOT_have act_ord_b: ‹❙𝒜[O!]b›
5900 using b_ord "≡E"(1) "oa-facts:7" by blast
5901 AOT_have delta_ord_b: ‹❙Δ[O!]b›
5902 by (meson "≡⇩d⇩fI" b_ord "∨I"(1) necessary_or_contingently_false "oa-facts:1" "vdash-properties:10")
5903 AOT_have not_act_ord_a: ‹¬❙𝒜[O!]a›
5904 by (meson a_prop "≡E"(1) "≡E"(3) "oa-contingent:3" "oa-facts:7")
5905 AOT_have not_delta_ord_a: ‹¬❙Δ[O!]a›
5906 by (metis Delta_pos "≡E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7" "reductio-aa:1" "vdash-properties:10")
5907
5908 AOT_have not_act_abs_b: ‹¬❙𝒜[A!]b›
5909 by (meson b_ord "≡E"(1) "≡E"(3) "oa-contingent:2" "oa-facts:8")
5910 AOT_have not_delta_abs_b: ‹¬❙Δ[A!]b›
5911 proof(rule "raa-cor:2")
5912 AOT_assume ‹❙Δ[A!]b›
5913 AOT_hence ‹◇[A!]b›
5914 by (metis Delta_pos "vdash-properties:10")
5915 AOT_thus ‹[A!]b & ¬[A!]b›
5916 by (metis b_ord "&I" "≡E"(1) "oa-contingent:2" "oa-facts:4" "vdash-properties:10")
5917 qed
5918 AOT_have act_abs_a: ‹❙𝒜[A!]a›
5919 using a_prop "≡E"(1) "oa-facts:8" by blast
5920 AOT_have delta_abs_a: ‹❙Δ[A!]a›
5921 by (metis "≡⇩d⇩fI" a_prop "oa-facts:2" "vdash-properties:10" "∨I"(1) necessary_or_contingently_false)
5922
5923 AOT_have not_act_concrete_b: ‹¬❙𝒜[E!]b›
5924 using b_prop "&E"(2) by blast
5925 AOT_have delta_concrete_b: ‹❙Δ[E!]b›
5926 proof (rule "≡⇩d⇩fI"[OF necessary_or_contingently_false]; rule "∨I"(2); rule "&I")
5927 AOT_show ‹¬❙𝒜[E!]b› using b_prop "&E"(2) by blast
5928 next
5929 AOT_show ‹◇[E!]b› using b_prop "&E"(1) by blast
5930 qed
5931 AOT_have not_act_concrete_a: ‹¬❙𝒜[E!]a›
5932 proof (rule "raa-cor:2")
5933 AOT_assume ‹❙𝒜[E!]a›
5934 AOT_hence 1: ‹◇[E!]a› by (metis "Act-Sub:3" "vdash-properties:10")
5935 AOT_have ‹[A!]a› by (simp add: a_prop)
5936 AOT_hence ‹[λx ¬◇[E!]x]a›
5937 by (rule "=⇩d⇩fE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5938 AOT_hence ‹¬◇[E!]a› using "β→C"(1) by blast
5939 AOT_thus ‹◇[E!]a & ¬◇[E!]a› using 1 "&I" by blast
5940 qed
5941 AOT_have not_delta_concrete_a: ‹¬❙Δ[E!]a›
5942 proof (rule "raa-cor:2")
5943 AOT_assume ‹❙Δ[E!]a›
5944 AOT_hence 1: ‹◇[E!]a› by (metis Delta_pos "vdash-properties:10")
5945 AOT_have ‹[A!]a› by (simp add: a_prop)
5946 AOT_hence ‹[λx ¬◇[E!]x]a›
5947 by (rule "=⇩d⇩fE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
5948 AOT_hence ‹¬◇[E!]a› using "β→C"(1) by blast
5949 AOT_thus ‹◇[E!]a & ¬◇[E!]a› using 1 "&I" by blast
5950 qed
5951
5952 AOT_have not_act_q_zero: ‹¬❙𝒜q⇩0›
5953 by (meson "log-prop-prop:2" "pos-not-pna:1" q⇩0_def "reductio-aa:1" "rule-id-df:2:a[zero]")
5954 AOT_have delta_q_zero: ‹❙Δq⇩0›
5955 proof(rule "≡⇩d⇩fI"[OF necessary_or_contingently_false]; rule "∨I"(2); rule "&I")
5956 AOT_show ‹¬❙𝒜q⇩0› using not_act_q_zero.
5957 AOT_show ‹◇q⇩0› by (meson "&E"(1) q⇩0_prop)
5958 qed
5959 AOT_have act_not_q_zero: ‹❙𝒜¬q⇩0› using "Act-Basic:1" "∨E"(2) not_act_q_zero by blast
5960 AOT_have not_delta_not_q_zero: ‹¬❙Δ¬q⇩0›
5961 using "≡⇩d⇩fE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta "&E"(1) "∨E"(2) not_act_q_zero q⇩0_prop by blast
5962
5963 AOT_have ‹[L⇧-]↓› by (simp add: "rel-neg-T:3")
5964 moreover AOT_have ‹¬❙𝒜[L⇧-]b & ¬❙Δ[L⇧-]b & ¬❙𝒜[L⇧-]a & ¬❙Δ[L⇧-]a›
5965 proof (safe intro!: "&I")
5966 AOT_show ‹¬❙𝒜[L⇧-]b› by (meson "≡E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act" nec_not_L_neg "→E")
5967 AOT_show ‹¬❙Δ[L⇧-]b› by (meson Delta_pos "KBasic2:1" "≡E"(1) "modus-tollens:1" nec_not_L_neg)
5968 AOT_show ‹¬❙𝒜[L⇧-]a› by (meson "≡E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act" nec_not_L_neg "→E")
5969 AOT_show ‹¬❙Δ[L⇧-]a› using Delta_pos "KBasic2:1" "≡E"(1) "modus-tollens:1" nec_not_L_neg by blast
5970 qed
5971 ultimately AOT_obtain F⇩0 where ‹¬❙𝒜[F⇩0]b & ¬❙Δ[F⇩0]b & ¬❙𝒜[F⇩0]a & ¬❙Δ[F⇩0]a›
5972 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5973 AOT_hence ‹¬❙𝒜[F⇩0]b› and ‹¬❙Δ[F⇩0]b› and ‹¬❙𝒜[F⇩0]a› and ‹¬❙Δ[F⇩0]a›
5974 using "&E" by blast+
5975 note props = this
5976
5977 let ?Π = "«[λy [A!]y & q⇩0]»"
5978 AOT_modally_strict {
5979 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
5980 } note 1 = this
5981 moreover AOT_have‹¬❙𝒜[«?Π»]b & ¬❙Δ[«?Π»]b & ¬❙𝒜[«?Π»]a & ❙Δ[«?Π»]a›
5982 proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
5983 AOT_show ‹¬❙𝒜([A!]b & q⇩0)›
5984 using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_abs_b "raa-cor:3" by blast
5985 next AOT_show ‹¬❙Δ([A!]b & q⇩0)›
5986 by (metis Delta_pos "KBasic2:3" "&E"(1) "≡E"(4) not_act_abs_b "oa-facts:4" "oa-facts:8" "raa-cor:3" "vdash-properties:10")
5987 next AOT_show ‹¬❙𝒜([A!]a & q⇩0)›
5988 using "Act-Basic:2" "&E"(2) "≡E"(1) not_act_q_zero "raa-cor:3" by blast
5989 next AOT_show ‹❙Δ([A!]a & q⇩0)›
5990 proof (rule not_act_and_pos_delta)
5991 AOT_show ‹¬❙𝒜([A!]a & q⇩0)›
5992 using "Act-Basic:2" "&E"(2) "≡E"(4) not_act_q_zero "raa-cor:3" by blast
5993 next AOT_show ‹◇([A!]a & q⇩0)›
5994 by (metis "&I" "→E" Delta_pos "KBasic:16" "&E"(1) delta_abs_a "≡E"(1) "oa-facts:6" q⇩0_prop)
5995 qed
5996 qed
5997 ultimately AOT_obtain F⇩1 where ‹¬❙𝒜[F⇩1]b & ¬❙Δ[F⇩1]b & ¬❙𝒜[F⇩1]a & ❙Δ[F⇩1]a›
5998 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
5999 AOT_hence ‹¬❙𝒜[F⇩1]b› and ‹¬❙Δ[F⇩1]b› and ‹¬❙𝒜[F⇩1]a› and ‹❙Δ[F⇩1]a›
6000 using "&E" by blast+
6001 note props = props this
6002
6003 let ?Π = "«[λy [A!]y & ¬q⇩0]»"
6004 AOT_modally_strict {
6005 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6006 } note 1 = this
6007 moreover AOT_have ‹¬❙𝒜[«?Π»]b & ¬❙Δ[«?Π»]b & ❙𝒜[«?Π»]a & ¬❙Δ[«?Π»]a›
6008 proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6009 AOT_show ‹¬❙𝒜([A!]b & ¬q⇩0)›
6010 using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_abs_b "raa-cor:3" by blast
6011 next AOT_show ‹¬❙Δ([A!]b & ¬q⇩0)›
6012 by (meson "RM◇" Delta_pos "Conjunction Simplification"(1) "≡E"(4) "modus-tollens:1" not_act_abs_b "oa-facts:4" "oa-facts:8")
6013 next AOT_show ‹❙𝒜([A!]a & ¬q⇩0)›
6014 by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:3")
6015 next AOT_show ‹¬❙Δ([A!]a & ¬q⇩0)›
6016 proof (rule act_and_not_nec_not_delta)
6017 AOT_show ‹❙𝒜([A!]a & ¬q⇩0)›
6018 by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:3")
6019 next
6020 AOT_show ‹¬□([A!]a & ¬q⇩0)›
6021 by (metis "KBasic2:1" "KBasic:3" "&E"(1) "&E"(2) "≡E"(4) q⇩0_prop "raa-cor:3")
6022 qed
6023 qed
6024 ultimately AOT_obtain F⇩2 where ‹¬❙𝒜[F⇩2]b & ¬❙Δ[F⇩2]b & ❙𝒜[F⇩2]a & ¬❙Δ[F⇩2]a›
6025 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6026 AOT_hence ‹¬❙𝒜[F⇩2]b› and ‹¬❙Δ[F⇩2]b› and ‹❙𝒜[F⇩2]a› and ‹¬❙Δ[F⇩2]a›
6027 using "&E" by blast+
6028 note props = props this
6029
6030 AOT_have abstract_prop: ‹¬❙𝒜[A!]b & ¬❙Δ[A!]b & ❙𝒜[A!]a & ❙Δ[A!]a›
6031 using act_abs_a "&I" delta_abs_a not_act_abs_b not_delta_abs_b by presburger
6032 then AOT_obtain F⇩3 where ‹¬❙𝒜[F⇩3]b & ¬❙Δ[F⇩3]b & ❙𝒜[F⇩3]a & ❙Δ[F⇩3]a›
6033 using "∃I"(1)[rotated, THEN "∃E"[rotated]] "oa-exist:2" by fastforce
6034 AOT_hence ‹¬❙𝒜[F⇩3]b› and ‹¬❙Δ[F⇩3]b› and ‹❙𝒜[F⇩3]a› and ‹❙Δ[F⇩3]a›
6035 using "&E" by blast+
6036 note props = props this
6037
6038 AOT_have ‹¬❙𝒜[E!]b & ❙Δ[E!]b & ¬❙𝒜[E!]a & ¬❙Δ[E!]a›
6039 by (meson "&I" delta_concrete_b not_act_concrete_a not_act_concrete_b not_delta_concrete_a)
6040 then AOT_obtain F⇩4 where ‹¬❙𝒜[F⇩4]b & ❙Δ[F⇩4]b & ¬❙𝒜[F⇩4]a & ¬❙Δ[F⇩4]a›
6041 using "cqt:2[concrete]"[axiom_inst] "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6042 AOT_hence ‹¬❙𝒜[F⇩4]b› and ‹❙Δ[F⇩4]b› and ‹¬❙𝒜[F⇩4]a› and ‹¬❙Δ[F⇩4]a›
6043 using "&E" by blast+
6044 note props = props this
6045
6046 AOT_modally_strict {
6047 AOT_have ‹[λy q⇩0]↓› by "cqt:2[lambda]"
6048 } note 1 = this
6049 moreover AOT_have ‹¬❙𝒜[λy q⇩0]b & ❙Δ[λy q⇩0]b & ¬❙𝒜[λy q⇩0]a & ❙Δ[λy q⇩0]a›
6050 by (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6051 (auto simp: not_act_q_zero delta_q_zero)
6052 ultimately AOT_obtain F⇩5 where ‹¬❙𝒜[F⇩5]b & ❙Δ[F⇩5]b & ¬❙𝒜[F⇩5]a & ❙Δ[F⇩5]a›
6053 using "cqt:2[concrete]"[axiom_inst] "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6054 AOT_hence ‹¬❙𝒜[F⇩5]b› and ‹❙Δ[F⇩5]b› and ‹¬❙𝒜[F⇩5]a› and ‹❙Δ[F⇩5]a›
6055 using "&E" by blast+
6056 note props = props this
6057
6058 let ?Π = "«[λy [E!]y ∨ ([A!]y & ¬q⇩0)]»"
6059 AOT_modally_strict {
6060 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6061 } note 1 = this
6062 moreover AOT_have ‹¬❙𝒜[«?Π»]b & ❙Δ[«?Π»]b & ❙𝒜[«?Π»]a & ¬❙Δ[«?Π»]a›
6063 proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6064 AOT_have ‹❙𝒜¬([A!]b & ¬q⇩0)›
6065 by (metis "Act-Basic:1" "Act-Basic:2" abstract_prop "&E"(1) "∨E"(2)
6066 "≡E"(1) "raa-cor:3")
6067 moreover AOT_have ‹¬❙𝒜[E!]b›
6068 using b_prop "&E"(2) by blast
6069 ultimately AOT_have 2: ‹❙𝒜(¬[E!]b & ¬([A!]b & ¬q⇩0))›
6070 by (metis "Act-Basic:2" "Act-Sub:1" "&I" "≡E"(3) "raa-cor:1")
6071 AOT_have ‹❙𝒜¬([E!]b ∨ ([A!]b & ¬q⇩0))›
6072 by (AOT_subst ‹¬([E!]b ∨ ([A!]b & ¬q⇩0))› ‹¬[E!]b & ¬([A!]b & ¬q⇩0)›)
6073 (auto simp: "oth-class-taut:5:d" 2)
6074 AOT_thus ‹¬❙𝒜([E!]b ∨ ([A!]b & ¬q⇩0))›
6075 by (metis "¬¬I" "Act-Sub:1" "≡E"(4))
6076 next
6077 AOT_show ‹❙Δ([E!]b ∨ ([A!]b & ¬q⇩0))›
6078 proof (rule not_act_and_pos_delta)
6079 AOT_show ‹¬❙𝒜([E!]b ∨ ([A!]b & ¬q⇩0))›
6080 by (metis "Act-Basic:2" "Act-Basic:9" "∨E"(2) "Conjunction Simplification"(1) "≡E"(4) "modus-tollens:1" not_act_abs_b not_act_concrete_b "raa-cor:3")
6081 next
6082 AOT_show ‹◇([E!]b ∨ ([A!]b & ¬q⇩0))›
6083 using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3) "raa-cor:3" by blast
6084 qed
6085 next AOT_show ‹❙𝒜([E!]a ∨ ([A!]a & ¬q⇩0))›
6086 by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_abs_a "&I" "∨I"(2) "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:1")
6087 next AOT_show ‹¬❙Δ([E!]a ∨ ([A!]a & ¬q⇩0))›
6088 proof (rule act_and_not_nec_not_delta)
6089 AOT_show ‹❙𝒜([E!]a ∨ ([A!]a & ¬q⇩0))›
6090 by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_abs_a "&I" "∨I"(2) "∨E"(2) "≡E"(3) not_act_q_zero "raa-cor:1")
6091 next
6092 AOT_have ‹□¬[E!]a›
6093 by (metis "≡⇩d⇩fI" "conventions:5" "&I" "∨I"(2) necessary_or_contingently_false not_act_concrete_a not_delta_concrete_a "raa-cor:3")
6094 moreover AOT_have ‹◇¬([A!]a & ¬q⇩0)›
6095 by (metis "KBasic2:1" "KBasic:11" "KBasic:3" "&E"(1) "&E"(2) "≡E"(1) q⇩0_prop "raa-cor:3")
6096 ultimately AOT_have ‹◇(¬[E!]a & ¬([A!]a & ¬q⇩0))› by (metis "KBasic:16" "&I" "vdash-properties:10")
6097 AOT_hence ‹◇¬([E!]a ∨ ([A!]a & ¬q⇩0))›
6098 by (metis "RE◇" "≡E"(2) "oth-class-taut:5:d")
6099 AOT_thus ‹¬□([E!]a ∨ ([A!]a & ¬q⇩0))› by (metis "KBasic:12" "≡E"(1) "raa-cor:3")
6100 qed
6101 qed
6102 ultimately AOT_obtain F⇩6 where ‹¬❙𝒜[F⇩6]b & ❙Δ[F⇩6]b & ❙𝒜[F⇩6]a & ¬❙Δ[F⇩6]a›
6103 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6104 AOT_hence ‹¬❙𝒜[F⇩6]b› and ‹❙Δ[F⇩6]b› and ‹❙𝒜[F⇩6]a› and ‹¬❙Δ[F⇩6]a›
6105 using "&E" by blast+
6106 note props = props this
6107
6108 let ?Π = "«[λy [A!]y ∨ [E!]y]»"
6109 AOT_modally_strict {
6110 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6111 } note 1 = this
6112 moreover AOT_have ‹¬❙𝒜[«?Π»]b & ❙Δ[«?Π»]b & ❙𝒜[«?Π»]a & ❙Δ[«?Π»]a›
6113 proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6114 AOT_show ‹¬❙𝒜([A!]b ∨ [E!]b)›
6115 using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b not_act_concrete_b "raa-cor:3" by blast
6116 next AOT_show ‹❙Δ([A!]b ∨ [E!]b)›
6117 proof (rule not_act_and_pos_delta)
6118 AOT_show ‹¬❙𝒜([A!]b ∨ [E!]b)›
6119 using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b not_act_concrete_b "raa-cor:3" by blast
6120 next AOT_show ‹◇([A!]b ∨ [E!]b)›
6121 using "KBasic2:2" b_prop "&E"(1) "∨I"(2) "≡E"(2) by blast
6122 qed
6123 next AOT_show ‹❙𝒜([A!]a ∨ [E!]a)›
6124 by (meson "Act-Basic:9" act_abs_a "∨I"(1) "≡E"(2))
6125 next AOT_show ‹❙Δ([A!]a ∨ [E!]a) ›
6126 proof (rule nec_delta)
6127 AOT_show ‹□([A!]a ∨ [E!]a)›
6128 by (metis "KBasic:15" act_abs_a act_and_not_nec_not_delta "Disjunction Addition"(1) delta_abs_a "raa-cor:3" "vdash-properties:10")
6129 qed
6130 qed
6131 ultimately AOT_obtain F⇩7 where ‹¬❙𝒜[F⇩7]b & ❙Δ[F⇩7]b & ❙𝒜[F⇩7]a & ❙Δ[F⇩7]a›
6132 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6133 AOT_hence ‹¬❙𝒜[F⇩7]b› and ‹❙Δ[F⇩7]b› and ‹❙𝒜[F⇩7]a› and ‹❙Δ[F⇩7]a›
6134 using "&E" by blast+
6135 note props = props this
6136
6137 let ?Π = "«[λy [O!]y & ¬[E!]y]»"
6138 AOT_modally_strict {
6139 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6140 } note 1 = this
6141 moreover AOT_have ‹❙𝒜[«?Π»]b & ¬❙Δ[«?Π»]b & ¬❙𝒜[«?Π»]a & ¬❙Δ[«?Π»]a›
6142 proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6143 AOT_show ‹❙𝒜([O!]b & ¬[E!]b)›
6144 by (metis "Act-Basic:1" "Act-Basic:2" act_ord_b "&I" "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:3")
6145 next AOT_show ‹¬❙Δ([O!]b & ¬[E!]b)›
6146 by (metis (no_types, hide_lams) "conventions:5" "Act-Sub:1" "RM:1" act_and_not_nec_not_delta "act-conj-act:3"
6147 act_ord_b b_prop "&I" "&E"(1) "Conjunction Simplification"(2) "df-rules-formulas[3]"
6148 "≡E"(3) "raa-cor:1" "→E")
6149 next AOT_show ‹¬❙𝒜([O!]a & ¬[E!]a)›
6150 using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_ord_a "raa-cor:3" by blast
6151 next AOT_have ‹¬◇([O!]a & ¬[E!]a)›
6152 by (metis "KBasic2:3" "&E"(1) "≡E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7" "raa-cor:3" "vdash-properties:10")
6153 AOT_thus ‹¬❙Δ([O!]a & ¬[E!]a)›
6154 by (rule impossible_delta)
6155 qed
6156 ultimately AOT_obtain F⇩8 where ‹❙𝒜[F⇩8]b & ¬❙Δ[F⇩8]b & ¬❙𝒜[F⇩8]a & ¬❙Δ[F⇩8]a›
6157 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6158 AOT_hence ‹❙𝒜[F⇩8]b› and ‹¬❙Δ[F⇩8]b› and ‹¬❙𝒜[F⇩8]a› and ‹¬❙Δ[F⇩8]a›
6159 using "&E" by blast+
6160 note props = props this
6161
6162
6163 let ?Π = "«[λy ¬[E!]y & ([O!]y ∨ q⇩0)]»"
6164 AOT_modally_strict {
6165 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6166 } note 1 = this
6167 moreover AOT_have ‹❙𝒜[«?Π»]b & ¬❙Δ[«?Π»]b & ¬❙𝒜[«?Π»]a & ❙Δ[«?Π»]a›
6168 proof(safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6169 AOT_show ‹❙𝒜(¬[E!]b & ([O!]b ∨ q⇩0))›
6170 by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I" "∨I"(1)
6171 "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
6172 next AOT_show ‹¬❙Δ(¬[E!]b & ([O!]b ∨ q⇩0))›
6173 proof (rule act_and_pos_not_not_delta)
6174 AOT_show ‹❙𝒜(¬[E!]b & ([O!]b ∨ q⇩0))›
6175 by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I" "∨I"(1)
6176 "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
6177 next
6178 AOT_show ‹◇¬(¬[E!]b & ([O!]b ∨ q⇩0))›
6179 proof (AOT_subst ‹¬(¬[E!]b & ([O!]b ∨ q⇩0))› ‹[E!]b ∨ ¬([O!]b ∨ q⇩0)›)
6180 AOT_modally_strict {
6181 AOT_show ‹¬(¬[E!]b & ([O!]b ∨ q⇩0)) ≡ [E!]b ∨ ¬([O!]b ∨ q⇩0)›
6182 by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2) "∨E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
6183 }
6184 next
6185 AOT_show ‹◇([E!]b ∨ ¬([O!]b ∨ q⇩0))›
6186 using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3) "raa-cor:3" by blast
6187 qed
6188 qed
6189 next
6190 AOT_show ‹¬❙𝒜(¬[E!]a & ([O!]a ∨ q⇩0))›
6191 using "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1) not_act_ord_a not_act_q_zero "reductio-aa:2" by blast
6192 next
6193 AOT_show ‹❙Δ(¬[E!]a & ([O!]a ∨ q⇩0))›
6194 proof (rule not_act_and_pos_delta)
6195 AOT_show ‹¬❙𝒜(¬[E!]a & ([O!]a ∨ q⇩0))›
6196 by (metis "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1) not_act_ord_a not_act_q_zero "reductio-aa:2")
6197 next
6198 AOT_have ‹□¬[E!]a›
6199 using "KBasic2:1" "≡E"(2) not_act_and_pos_delta not_act_concrete_a not_delta_concrete_a "raa-cor:5" by blast
6200 moreover AOT_have ‹◇([O!]a ∨ q⇩0)›
6201 by (metis "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(3) q⇩0_prop "raa-cor:3")
6202 ultimately AOT_show ‹◇(¬[E!]a & ([O!]a ∨ q⇩0))›
6203 by (metis "KBasic:16" "&I" "vdash-properties:10")
6204 qed
6205 qed
6206 ultimately AOT_obtain F⇩9 where ‹❙𝒜[F⇩9]b & ¬❙Δ[F⇩9]b & ¬❙𝒜[F⇩9]a & ❙Δ[F⇩9]a›
6207 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6208 AOT_hence ‹❙𝒜[F⇩9]b› and ‹¬❙Δ[F⇩9]b› and ‹¬❙𝒜[F⇩9]a› and ‹❙Δ[F⇩9]a›
6209 using "&E" by blast+
6210 note props = props this
6211
6212 AOT_modally_strict {
6213 AOT_have ‹[λy ¬q⇩0]↓› by "cqt:2[lambda]"
6214 } note 1 = this
6215 moreover AOT_have ‹❙𝒜[λy ¬q⇩0]b & ¬❙Δ[λy ¬q⇩0]b & ❙𝒜[λy ¬q⇩0]a & ¬❙Δ[λy ¬q⇩0]a›
6216 by (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1]; auto simp: act_not_q_zero not_delta_not_q_zero)
6217 ultimately AOT_obtain F⇩1⇩0 where ‹❙𝒜[F⇩1⇩0]b & ¬❙Δ[F⇩1⇩0]b & ❙𝒜[F⇩1⇩0]a & ¬❙Δ[F⇩1⇩0]a›
6218 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6219 AOT_hence ‹❙𝒜[F⇩1⇩0]b› and ‹¬❙Δ[F⇩1⇩0]b› and ‹❙𝒜[F⇩1⇩0]a› and ‹¬❙Δ[F⇩1⇩0]a›
6220 using "&E" by blast+
6221 note props = props this
6222
6223 AOT_modally_strict {
6224 AOT_have ‹[λy ¬[E!]y]↓› by "cqt:2[lambda]"
6225 } note 1 = this
6226 moreover AOT_have ‹❙𝒜[λy ¬[E!]y]b & ¬❙Δ[λy ¬[E!]y]b & ❙𝒜[λy ¬[E!]y]a & ❙Δ[λy ¬[E!]y]a›
6227 proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6228 AOT_show ‹❙𝒜¬[E!]b›
6229 using "Act-Basic:1" "∨E"(2) not_act_concrete_b by blast
6230 next AOT_show ‹¬❙Δ¬[E!]b›
6231 using "≡⇩d⇩fE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta b_prop "&E"(1) "∨E"(2) not_act_concrete_b by blast
6232 next AOT_show ‹❙𝒜¬[E!]a›
6233 using "Act-Basic:1" "∨E"(2) not_act_concrete_a by blast
6234 next AOT_show ‹❙Δ¬[E!]a›
6235 using "KBasic2:1" "≡E"(2) nec_delta not_act_and_pos_delta not_act_concrete_a not_delta_concrete_a "reductio-aa:1" by blast
6236 qed
6237 ultimately AOT_obtain F⇩1⇩1 where ‹❙𝒜[F⇩1⇩1]b & ¬❙Δ[F⇩1⇩1]b & ❙𝒜[F⇩1⇩1]a & ❙Δ[F⇩1⇩1]a›
6238 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6239 AOT_hence ‹❙𝒜[F⇩1⇩1]b› and ‹¬❙Δ[F⇩1⇩1]b› and ‹❙𝒜[F⇩1⇩1]a› and ‹❙Δ[F⇩1⇩1]a›
6240 using "&E" by blast+
6241 note props = props this
6242
6243 AOT_have ‹❙𝒜[O!]b & ❙Δ[O!]b & ¬❙𝒜[O!]a & ¬❙Δ[O!]a›
6244 by (simp add: act_ord_b "&I" delta_ord_b not_act_ord_a not_delta_ord_a)
6245 then AOT_obtain F⇩1⇩2 where ‹❙𝒜[F⇩1⇩2]b & ❙Δ[F⇩1⇩2]b & ¬❙𝒜[F⇩1⇩2]a & ¬❙Δ[F⇩1⇩2]a›
6246 using "oa-exist:1" "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6247 AOT_hence ‹❙𝒜[F⇩1⇩2]b› and ‹❙Δ[F⇩1⇩2]b› and ‹¬❙𝒜[F⇩1⇩2]a› and ‹¬❙Δ[F⇩1⇩2]a›
6248 using "&E" by blast+
6249 note props = props this
6250
6251 let ?Π = "«[λy [O!]y ∨ q⇩0]»"
6252 AOT_modally_strict {
6253 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6254 } note 1 = this
6255 moreover AOT_have ‹❙𝒜[«?Π»]b & ❙Δ[«?Π»]b & ¬❙𝒜[«?Π»]a & ❙Δ[«?Π»]a›
6256 proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6257 AOT_show ‹❙𝒜([O!]b ∨ q⇩0)›
6258 by (meson "Act-Basic:9" act_ord_b "∨I"(1) "≡E"(2))
6259 next AOT_show ‹❙Δ([O!]b ∨ q⇩0)›
6260 by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "vdash-properties:10")
6261 next AOT_show ‹¬❙𝒜([O!]a ∨ q⇩0)›
6262 using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a not_act_q_zero "raa-cor:3" by blast
6263 next AOT_show ‹❙Δ([O!]a ∨ q⇩0)›
6264 proof (rule not_act_and_pos_delta)
6265 AOT_show ‹¬❙𝒜([O!]a ∨ q⇩0)›
6266 using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a not_act_q_zero "raa-cor:3" by blast
6267 next AOT_show ‹◇([O!]a ∨ q⇩0)›
6268 using "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(2) q⇩0_prop by blast
6269 qed
6270 qed
6271 ultimately AOT_obtain F⇩1⇩3 where ‹❙𝒜[F⇩1⇩3]b & ❙Δ[F⇩1⇩3]b & ¬❙𝒜[F⇩1⇩3]a & ❙Δ[F⇩1⇩3]a›
6272 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6273 AOT_hence ‹❙𝒜[F⇩1⇩3]b› and ‹❙Δ[F⇩1⇩3]b› and ‹¬❙𝒜[F⇩1⇩3]a› and ‹❙Δ[F⇩1⇩3]a›
6274 using "&E" by blast+
6275 note props = props this
6276
6277 let ?Π = "«[λy [O!]y ∨ ¬q⇩0]»"
6278 AOT_modally_strict {
6279 AOT_have ‹[«?Π»]↓› by "cqt:2[lambda]"
6280 } note 1 = this
6281 moreover AOT_have ‹❙𝒜[«?Π»]b & ❙Δ[«?Π»]b & ❙𝒜[«?Π»]a & ¬❙Δ[«?Π»]a›
6282 proof (safe intro!: "&I"; AOT_subst_using subst: "beta-C-meta"[THEN "→E", OF 1])
6283 AOT_show ‹❙𝒜([O!]b ∨ ¬q⇩0)›
6284 by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6285 next AOT_show ‹❙Δ([O!]b ∨ ¬q⇩0)›
6286 by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "vdash-properties:10")
6287 next AOT_show ‹❙𝒜([O!]a ∨ ¬q⇩0)›
6288 by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6289 next AOT_show ‹¬❙Δ([O!]a ∨ ¬q⇩0)›
6290 proof(rule act_and_pos_not_not_delta)
6291 AOT_show ‹❙𝒜([O!]a ∨ ¬q⇩0)›
6292 by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
6293 next
6294 AOT_have ‹□¬[O!]a›
6295 using "KBasic2:1" "≡E"(2) not_act_and_pos_delta not_act_ord_a not_delta_ord_a "raa-cor:6" by blast
6296 moreover AOT_have ‹◇q⇩0›
6297 by (meson "&E"(1) q⇩0_prop)
6298 ultimately AOT_have 2: ‹◇(¬[O!]a & q⇩0)›
6299 by (metis "KBasic:16" "&I" "vdash-properties:10")
6300 AOT_show ‹◇¬([O!]a ∨ ¬q⇩0)›
6301 proof (AOT_subst (reverse) ‹¬([O!]a ∨ ¬q⇩0)› ‹¬[O!]a & q⇩0›)
6302 AOT_modally_strict {
6303 AOT_show ‹¬[O!]a & q⇩0 ≡ ¬([O!]a ∨ ¬q⇩0)›
6304 by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2)
6305 "∨E"(3) "deduction-theorem" "≡I" "raa-cor:3")
6306 }
6307 next
6308 AOT_show ‹◇(¬[O!]a & q⇩0)›
6309 using "2" by blast
6310 qed
6311 qed
6312 qed
6313 ultimately AOT_obtain F⇩1⇩4 where ‹❙𝒜[F⇩1⇩4]b & ❙Δ[F⇩1⇩4]b & ❙𝒜[F⇩1⇩4]a & ¬❙Δ[F⇩1⇩4]a›
6314 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6315 AOT_hence ‹❙𝒜[F⇩1⇩4]b› and ‹❙Δ[F⇩1⇩4]b› and ‹❙𝒜[F⇩1⇩4]a› and ‹¬❙Δ[F⇩1⇩4]a›
6316 using "&E" by blast+
6317 note props = props this
6318
6319 AOT_have ‹[L]↓›
6320 by (rule "=⇩d⇩fI"(2)[OF L_def]) "cqt:2[lambda]"+
6321 moreover AOT_have ‹❙𝒜[L]b & ❙Δ[L]b & ❙𝒜[L]a & ❙Δ[L]a›
6322 proof (safe intro!: "&I")
6323 AOT_show ‹❙𝒜[L]b›
6324 by (meson nec_L "nec-imp-act" "vdash-properties:10")
6325 next AOT_show ‹❙Δ[L]b› using nec_L nec_delta by blast
6326 next AOT_show ‹❙𝒜[L]a› by (meson nec_L "nec-imp-act" "vdash-properties:10")
6327 next AOT_show ‹❙Δ[L]a› using nec_L nec_delta by blast
6328 qed
6329 ultimately AOT_obtain F⇩1⇩5 where ‹❙𝒜[F⇩1⇩5]b & ❙Δ[F⇩1⇩5]b & ❙𝒜[F⇩1⇩5]a & ❙Δ[F⇩1⇩5]a›
6330 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
6331 AOT_hence ‹❙𝒜[F⇩1⇩5]b› and ‹❙Δ[F⇩1⇩5]b› and ‹❙𝒜[F⇩1⇩5]a› and ‹❙Δ[F⇩1⇩5]a›
6332 using "&E" by blast+
6333 note props = props this
6334
6335 show ?thesis
6336 by (rule "∃I"(2)[where β=F⇩0]; rule "∃I"(2)[where β=F⇩1]; rule "∃I"(2)[where β=F⇩2];
6337 rule "∃I"(2)[where β=F⇩3]; rule "∃I"(2)[where β=F⇩4]; rule "∃I"(2)[where β=F⇩5];
6338 rule "∃I"(2)[where β=F⇩6]; rule "∃I"(2)[where β=F⇩7]; rule "∃I"(2)[where β=F⇩8];
6339 rule "∃I"(2)[where β=F⇩9]; rule "∃I"(2)[where β=F⇩1⇩0]; rule "∃I"(2)[where β=F⇩1⇩1];
6340 rule "∃I"(2)[where β=F⇩1⇩2]; rule "∃I"(2)[where β=F⇩1⇩3]; rule "∃I"(2)[where β=F⇩1⇩4];
6341 rule "∃I"(2)[where β=F⇩1⇩5]; safe intro!: "&I")
6342 (match conclusion in "[?v ⊨ [F] ≠ [G]]" for F G ⇒ ‹
6343 match props in A: "[?v ⊨ ¬φ{F}]" for φ ⇒ ‹
6344 match (φ) in "λa . ?p" ⇒ ‹fail› ¦ "λa . a" ⇒ ‹fail› ¦ _ ⇒ ‹
6345 match props in B: "[?v ⊨ φ{G}]" ⇒ ‹
6346 fact "pos-not-equiv-ne:4"[where F=F and G=G and φ=φ, THEN "→E",
6347 OF "oth-class-taut:4:h"[THEN "≡E"(2)],
6348 OF "Disjunction Addition"(2)[THEN "→E"],
6349 OF "&I", OF A, OF B]››››)+
6350qed
6351
6352AOT_theorem "o-objects-exist:1": ‹□∃x O!x›
6353proof(rule RN)
6354 AOT_modally_strict {
6355 AOT_obtain a where ‹◇(E!a & ¬❙𝒜[E!]a)›
6356 using "∃E"[rotated, OF "qml:4"[axiom_inst, THEN "BF◇"[THEN "→E"]]] by blast
6357 AOT_hence 1: ‹◇E!a› by (metis "KBasic2:3" "&E"(1) "→E")
6358 AOT_have ‹[λx ◇[E!]x]a›
6359 proof (rule "β←C"(1); "cqt:2[lambda]"?)
6360 AOT_show ‹a↓› using "cqt:2[const_var]"[axiom_inst] by blast
6361 next
6362 AOT_show ‹◇E!a› by (fact 1)
6363 qed
6364 AOT_hence ‹O!a› by (rule "=⇩d⇩fI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
6365 AOT_thus ‹∃x [O!]x› by (rule "∃I")
6366 }
6367qed
6368
6369AOT_theorem "o-objects-exist:2": ‹□∃x A!x›
6370proof (rule RN)
6371 AOT_modally_strict {
6372 AOT_obtain a where ‹[A!]a›
6373 using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
6374 AOT_thus ‹∃x A!x› using "∃I" by blast
6375 }
6376qed
6377
6378AOT_theorem "o-objects-exist:3": ‹□¬∀x O!x›
6379 by (rule RN) (metis (no_types, hide_lams) "∃E" "cqt-orig:1[const_var]" "≡E"(4) "modus-tollens:1" "o-objects-exist:2" "oa-contingent:2" "qml:2"[axiom_inst] "reductio-aa:2")
6380
6381AOT_theorem "o-objects-exist:4": ‹□¬∀x A!x›
6382 by (rule RN) (metis (mono_tags, hide_lams) "∃E" "cqt-orig:1[const_var]" "≡E"(1) "modus-tollens:1" "o-objects-exist:1" "oa-contingent:2" "qml:2"[axiom_inst] "→E")
6383
6384AOT_theorem "o-objects-exist:5": ‹□¬∀x E!x›
6385proof (rule RN; rule "raa-cor:2")
6386 AOT_modally_strict {
6387 AOT_assume ‹∀x E!x›
6388 moreover AOT_obtain a where abs: ‹A!a›
6389 using "o-objects-exist:2"[THEN "qml:2"[axiom_inst, THEN "→E"]] "∃E"[rotated] by blast
6390 ultimately AOT_have ‹E!a› using "∀E" by blast
6391 AOT_hence 1: ‹◇E!a› by (metis "T◇" "→E")
6392 AOT_have ‹[λy ◇E!y]a›
6393 proof (rule "β←C"(1); "cqt:2[lambda]"?)
6394 AOT_show ‹a↓› using "cqt:2[const_var]"[axiom_inst].
6395 next
6396 AOT_show ‹◇E!a› by (fact 1)
6397 qed
6398 AOT_hence ‹O!a›
6399 by (rule "=⇩d⇩fI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
6400 AOT_hence ‹¬A!a› by (metis "≡E"(1) "oa-contingent:2")
6401 AOT_thus ‹p & ¬p› for p using abs by (metis "raa-cor:3")
6402 }
6403qed
6404
6405AOT_theorem partition: ‹¬∃x (O!x & A!x)›
6406proof(rule "raa-cor:2")
6407 AOT_assume ‹∃x (O!x & A!x)›
6408 then AOT_obtain a where ‹O!a & A!a› using "∃E"[rotated] by blast
6409 AOT_thus ‹p & ¬p› for p by (metis "&E"(1) "Conjunction Simplification"(2) "≡E"(1) "modus-tollens:1" "oa-contingent:2" "raa-cor:3")
6410qed
6411
6412AOT_define eq_E :: ‹Π› ("'(=⇩E')") "=E": ‹(=⇩E) =⇩d⇩f [λxy O!x & O!y & □∀F ([F]x ≡ [F]y)]›
6413
6414syntax "_AOT_eq_E_infix" :: ‹τ ⇒ τ ⇒ φ› (infixl "=⇩E" 50)
6415translations
6416 "_AOT_eq_E_infix κ κ'" == "CONST AOT_exe (CONST eq_E) (CONST Pair κ κ')"
6417
6418print_translation‹
6419AOT_syntax_print_translations
6420[(\<^const_syntax>‹AOT_exe›, fn ctxt => fn [
6421 Const ("\<^const>AOT_PLM.eq_E", _),
6422 Const (\<^const_syntax>‹Pair›, _) $ lhs $ rhs
6423] => Const (\<^syntax_const>‹_AOT_eq_E_infix›, dummyT) $ lhs $ rhs)]›
6424
6425text‹Note: Not explicitly mentioned as theorem in PLM.›
6426AOT_theorem "=E[denotes]": ‹[(=⇩E)]↓›
6427 by (rule "=⇩d⇩fI"(2)[OF "=E"]) "cqt:2[lambda]"+
6428
6429AOT_theorem "=E-simple:1": ‹x =⇩E y ≡ (O!x & O!y & □∀F ([F]x ≡ [F]y))›
6430proof -
6431
6432 AOT_have 0: ‹«(AOT_term_of_var x,AOT_term_of_var y)»↓›
6433 by (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6434 AOT_have 1: ‹[λxy [O!]x & [O!]y & □∀F ([F]x ≡ [F]y)]↓› by "cqt:2[lambda]"
6435 show ?thesis apply (rule "=⇩d⇩fI"(2)[OF "=E"]; "cqt:2[lambda]"?)
6436 using "beta-C-meta"[THEN "→E", OF 1, unvarify ν⇩1ν⇩n, of "(AOT_term_of_var x,AOT_term_of_var y)", OF 0]
6437 by fast
6438qed
6439
6440AOT_theorem "=E-simple:2": ‹x =⇩E y → x = y›
6441proof (rule "→I")
6442 AOT_assume ‹x =⇩E y›
6443 AOT_hence ‹O!x & O!y & □∀F ([F]x ≡ [F]y)› using "=E-simple:1"[THEN "≡E"(1)] by blast
6444 AOT_thus ‹x = y›
6445 using "≡⇩d⇩fI"[OF "identity:1"] "∨I" by blast
6446qed
6447
6448AOT_theorem "id-nec3:1": ‹x =⇩E y ≡ □(x =⇩E y)›
6449proof (rule "≡I"; rule "→I")
6450 AOT_assume ‹x =⇩E y›
6451 AOT_hence ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›
6452 using "=E-simple:1" "≡E" by blast
6453 AOT_hence ‹□O!x & □O!y & □□∀F ([F]x ≡ [F]y)›
6454 by (metis "S5Basic:6" "&I" "&E"(1) "&E"(2) "≡E"(4) "oa-facts:1" "raa-cor:3" "vdash-properties:10")
6455 AOT_hence ‹□(O!x & O!y & □∀F ([F]x ≡ [F]y))›
6456 by (metis "&E"(1) "&E"(2) "≡E"(2) "KBasic:3" "&I")
6457 AOT_thus ‹□(x =⇩E y)›
6458 using "=E-simple:1"
6459 by (AOT_subst ‹x =⇩E y› ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›) auto
6460next
6461 AOT_assume ‹□(x =⇩E y)›
6462 AOT_thus ‹x =⇩E y› using "qml:2"[axiom_inst, THEN "→E"] by blast
6463qed
6464
6465AOT_theorem "id-nec3:2": ‹◇(x =⇩E y) ≡ x =⇩E y›
6466 by (meson "RE◇" "S5Basic:2" "id-nec3:1" "≡E"(1) "≡E"(5) "Commutativity of ≡")
6467
6468AOT_theorem "id-nec3:3": ‹◇(x =⇩E y) ≡ □(x =⇩E y)›
6469 by (meson "id-nec3:1" "id-nec3:2" "≡E"(5))
6470
6471syntax "_AOT_non_eq_E" :: ‹Π› ("'(≠⇩E')")
6472translations
6473 (Π) "(≠⇩E)" == (Π) "(=⇩E)⇧-"
6474syntax "_AOT_non_eq_E_infix" :: ‹τ ⇒ τ ⇒ φ› (infixl "≠⇩E" 50)
6475translations
6476 "_AOT_non_eq_E_infix κ κ'" == "CONST AOT_exe (CONST relation_negation (CONST eq_E)) (CONST Pair κ κ')"
6477
6478print_translation‹
6479AOT_syntax_print_translations
6480[(\<^const_syntax>‹AOT_exe›, fn ctxt => fn [
6481 Const (\<^const_syntax>‹relation_negation›, _) $ Const ("\<^const>AOT_PLM.eq_E", _),
6482 Const (\<^const_syntax>‹Pair›, _) $ lhs $ rhs
6483] => Const (\<^syntax_const>‹_AOT_non_eq_E_infix›, dummyT) $ lhs $ rhs)]›
6484AOT_theorem "thm-neg=E": ‹x ≠⇩E y ≡ ¬(x =⇩E y)›
6485proof -
6486
6487 AOT_have 0: ‹«(AOT_term_of_var x,AOT_term_of_var y)»↓›
6488 by (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6489 AOT_have θ: ‹[λx⇩1...x⇩2 ¬(=⇩E)x⇩1...x⇩2]↓› by "cqt:2[lambda]"
6490 AOT_have ‹x ≠⇩E y ≡ [λx⇩1...x⇩2 ¬(=⇩E)x⇩1...x⇩2]xy›
6491 by (rule "=⇩d⇩fI"(1)[OF "df-relation-negation", OF θ])
6492 (meson "oth-class-taut:3:a")
6493 also AOT_have ‹… ≡ ¬(=⇩E)xy›
6494 apply (rule "beta-C-meta"[THEN "→E", unvarify ν⇩1ν⇩n])
6495 apply "cqt:2[lambda]"
6496 by (fact 0)
6497 finally show ?thesis.
6498qed
6499
6500AOT_theorem "id-nec4:1": ‹x ≠⇩E y ≡ □(x ≠⇩E y)›
6501proof -
6502 AOT_have ‹x ≠⇩E y ≡ ¬(x =⇩E y)› using "thm-neg=E".
6503 also AOT_have ‹… ≡ ¬◇(x =⇩E y)›
6504 by (meson "id-nec3:2" "≡E"(1) "Commutativity of ≡" "oth-class-taut:4:b")
6505 also AOT_have ‹… ≡ □¬(x =⇩E y)›
6506 by (meson "KBasic2:1" "≡E"(2) "Commutativity of ≡")
6507 also AOT_have ‹… ≡ □(x ≠⇩E y)›
6508 by (AOT_subst (reverse) ‹¬(x =⇩E y)› ‹x ≠⇩E y›)
6509 (auto simp: "thm-neg=E" "oth-class-taut:3:a")
6510 finally show ?thesis.
6511qed
6512
6513AOT_theorem "id-nec4:2": ‹◇(x ≠⇩E y) ≡ (x ≠⇩E y)›
6514 by (meson "RE◇" "S5Basic:2" "id-nec4:1" "≡E"(2) "≡E"(5) "Commutativity of ≡")
6515
6516AOT_theorem "id-nec4:3": ‹◇(x ≠⇩E y) ≡ □(x ≠⇩E y)›
6517 by (meson "id-nec4:1" "id-nec4:2" "≡E"(5))
6518
6519AOT_theorem "id-act2:1": ‹x =⇩E y ≡ ❙𝒜x =⇩E y›
6520 by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec3:2" "≡E"(1) "≡E"(6))
6521AOT_theorem "id-act2:2": ‹x ≠⇩E y ≡ ❙𝒜x ≠⇩E y›
6522 by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec4:2" "≡E"(1) "≡E"(6))
6523
6524AOT_theorem "ord=Eequiv:1": ‹O!x → x =⇩E x›
6525proof (rule "→I")
6526 AOT_assume 1: ‹O!x›
6527 AOT_show ‹x =⇩E x›
6528 apply (rule "=⇩d⇩fI"(2)[OF "=E"]) apply "cqt:2[lambda]"
6529 apply (rule "β←C"(1))
6530 apply "cqt:2[lambda]"
6531 apply (simp add: "&I" "cqt:2[const_var]" prod_denotesI "vdash-properties:1[2]")
6532 by (simp add: "1" RN "&I" "oth-class-taut:3:a" "universal-cor")
6533qed
6534
6535AOT_theorem "ord=Eequiv:2": ‹x =⇩E y → y =⇩E x›
6536proof(rule CP)
6537 AOT_assume 1: ‹x =⇩E y›
6538 AOT_hence 2: ‹x = y› by (metis "=E-simple:2" "vdash-properties:10")
6539 AOT_have ‹O!x› using 1 by (meson "&E"(1) "=E-simple:1" "≡E"(1))
6540 AOT_hence ‹x =⇩E x› using "ord=Eequiv:1" "→E" by blast
6541 AOT_thus ‹y =⇩E x› using "rule=E"[rotated, OF 2] by fast
6542qed
6543
6544AOT_theorem "ord=Eequiv:3": ‹(x =⇩E y & y =⇩E z) → x =⇩E z›
6545proof (rule CP)
6546 AOT_assume 1: ‹x =⇩E y & y =⇩E z›
6547 AOT_hence ‹x = y & y = z›
6548 by (metis "&I" "&E"(1) "&E"(2) "=E-simple:2" "vdash-properties:6")
6549 AOT_hence ‹x = z› by (metis "id-eq:3" "vdash-properties:6")
6550 moreover AOT_have ‹x =⇩E x›
6551 using 1[THEN "&E"(1)] "&E"(1) "=E-simple:1" "≡E"(1) "ord=Eequiv:1" "→E" by blast
6552 ultimately AOT_show ‹x =⇩E z›
6553 using "rule=E" by fast
6554qed
6555
6556AOT_theorem "ord-=E=:1": ‹(O!x ∨ O!y) → □(x = y ≡ x =⇩E y)›
6557proof(rule CP)
6558 AOT_assume ‹O!x ∨ O!y›
6559 moreover {
6560 AOT_assume ‹O!x›
6561 AOT_hence ‹□O!x› by (metis "oa-facts:1" "vdash-properties:10")
6562 moreover {
6563 AOT_modally_strict {
6564 AOT_have ‹O!x → (x = y ≡ x =⇩E y)›
6565 proof (rule "→I"; rule "≡I"; rule "→I")
6566 AOT_assume ‹O!x›
6567 AOT_hence ‹x =⇩E x› by (metis "ord=Eequiv:1" "→E")
6568 moreover AOT_assume ‹x = y›
6569 ultimately AOT_show ‹x =⇩E y› using "rule=E" by fast
6570 next
6571 AOT_assume ‹x =⇩E y›
6572 AOT_thus ‹x = y› by (metis "=E-simple:2" "→E")
6573 qed
6574 }
6575 AOT_hence ‹□O!x → □(x = y ≡ x =⇩E y)› by (metis "RM:1")
6576 }
6577 ultimately AOT_have ‹□(x = y ≡ x =⇩E y)› using "→E" by blast
6578 }
6579 moreover {
6580 AOT_assume ‹O!y›
6581 AOT_hence ‹□O!y› by (metis "oa-facts:1" "vdash-properties:10")
6582 moreover {
6583 AOT_modally_strict {
6584 AOT_have ‹O!y → (x = y ≡ x =⇩E y)›
6585 proof (rule "→I"; rule "≡I"; rule "→I")
6586 AOT_assume ‹O!y›
6587 AOT_hence ‹y =⇩E y› by (metis "ord=Eequiv:1" "→E")
6588 moreover AOT_assume ‹x = y›
6589 ultimately AOT_show ‹x =⇩E y› using "rule=E" id_sym by fast
6590 next
6591 AOT_assume ‹x =⇩E y›
6592 AOT_thus ‹x = y› by (metis "=E-simple:2" "→E")
6593 qed
6594 }
6595 AOT_hence ‹□O!y → □(x = y ≡ x =⇩E y)› by (metis "RM:1")
6596 }
6597 ultimately AOT_have ‹□(x = y ≡ x =⇩E y)› using "→E" by blast
6598 }
6599 ultimately AOT_show ‹□(x = y ≡ x =⇩E y)› by (metis "∨E"(3) "raa-cor:1")
6600qed
6601
6602AOT_theorem "ord-=E=:2": ‹O!y → [λx x = y]↓›
6603proof (rule "→I"; rule "safe-ext"[axiom_inst, THEN "→E"]; rule "&I")
6604 AOT_show ‹[λx x =⇩E y]↓› by "cqt:2[lambda]"
6605next
6606 AOT_assume ‹O!y›
6607 AOT_hence 1: ‹□(x = y ≡ x =⇩E y)› for x using "ord-=E=:1" "→E" "∨I" by blast
6608 AOT_have ‹□(x =⇩E y ≡ x = y)› for x
6609 by (AOT_subst ‹x =⇩E y ≡ x = y› ‹x = y ≡ x =⇩E y›)
6610 (auto simp add: "Commutativity of ≡" 1)
6611 AOT_hence ‹∀x □(x =⇩E y ≡ x = y)› by (rule GEN)
6612 AOT_thus ‹□∀x (x =⇩E y ≡ x = y)› by (rule BF[THEN "→E"])
6613qed
6614
6615
6616AOT_theorem "ord-=E=:3": ‹[λxy O!x & O!y & x = y]↓›
6617proof (rule "safe-ext[2]"[axiom_inst, THEN "→E"]; rule "&I")
6618 AOT_show ‹[λxy O!x & O!y & x =⇩E y]↓› by "cqt:2[lambda]"
6619next
6620 AOT_show ‹□∀x∀y ([O!]x & [O!]y & x =⇩E y ≡ [O!]x & [O!]y & x = y)›
6621 proof (rule RN; rule GEN; rule GEN; rule "≡I"; rule "→I")
6622 AOT_modally_strict {
6623 AOT_show ‹[O!]x & [O!]y & x = y› if ‹[O!]x & [O!]y & x =⇩E y› for x y
6624 by (metis "&I" "&E"(1) "Conjunction Simplification"(2) "=E-simple:2"
6625 "modus-tollens:1" "raa-cor:1" that)
6626 }
6627 next
6628 AOT_modally_strict {
6629 AOT_show ‹[O!]x & [O!]y & x =⇩E y› if ‹[O!]x & [O!]y & x = y› for x y
6630 apply(safe intro!: "&I")
6631 apply (metis that[THEN "&E"(1), THEN "&E"(1)])
6632 apply (metis that[THEN "&E"(1), THEN "&E"(2)])
6633 using "rule=E"[rotated, OF that[THEN "&E"(2)]]
6634 "ord=Eequiv:1"[THEN "→E", OF that[THEN "&E"(1), THEN "&E"(1)]] by fast
6635 }
6636 qed
6637qed
6638
6639AOT_theorem "ind-nec": ‹∀F ([F]x ≡ [F]y) → □∀F ([F]x ≡ [F]y)›
6640proof(rule "→I")
6641 AOT_assume ‹∀F ([F]x ≡ [F]y)›
6642 moreover AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]↓› by "cqt:2[lambda]"
6643 ultimately AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]x ≡ [λx □∀F ([F]x ≡ [F]y)]y›
6644 using "∀E" by blast
6645 moreover AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]y›
6646 apply (rule "β←C"(1))
6647 apply "cqt:2[lambda]"
6648 apply (fact "cqt:2[const_var]"[axiom_inst])
6649 by (simp add: RN GEN "oth-class-taut:3:a")
6650 ultimately AOT_have ‹[λx □∀F ([F]x ≡ [F]y)]x› using "≡E" by blast
6651 AOT_thus ‹□∀F ([F]x ≡ [F]y)›
6652 using "β→C"(1) by blast
6653qed
6654
6655AOT_theorem "ord=E:1": ‹(O!x & O!y) → (∀F ([F]x ≡ [F]y) → x =⇩E y)›
6656proof (rule "→I"; rule "→I")
6657 AOT_assume ‹∀F ([F]x ≡ [F]y)›
6658 AOT_hence ‹□∀F ([F]x ≡ [F]y)›
6659 using "ind-nec"[THEN "→E"] by blast
6660 moreover AOT_assume ‹O!x & O!y›
6661 ultimately AOT_have ‹O!x & O!y & □∀F ([F]x ≡ [F]y)›
6662 using "&I" by blast
6663 AOT_thus ‹x =⇩E y› using "=E-simple:1"[THEN "≡E"(2)] by blast
6664qed
6665
6666AOT_theorem "ord=E:2": ‹(O!x & O!y) → (∀F ([F]x ≡ [F]y) → x = y)›
6667proof (rule "→I"; rule "→I")
6668 AOT_assume ‹O!x & O!y›
6669 moreover AOT_assume ‹∀F ([F]x ≡ [F]y)›
6670 ultimately AOT_have ‹x =⇩E y›
6671 using "ord=E:1" "→E" by blast
6672 AOT_thus ‹x = y› using "=E-simple:2"[THEN "→E"] by blast
6673qed
6674
6675AOT_theorem "ord=E2:1": ‹(O!x & O!y) → (x ≠ y ≡ [λz z =⇩E x] ≠ [λz z =⇩E y])›
6676proof (rule "→I"; rule "≡I"; rule "→I"; rule "≡⇩d⇩fI"[OF "=-infix"]; rule "raa-cor:2")
6677 AOT_assume 0: ‹O!x & O!y›
6678 AOT_assume ‹x ≠ y›
6679 AOT_hence 1: ‹¬(x = y)› using "≡⇩d⇩fE"[OF "=-infix"] by blast
6680 AOT_assume ‹[λz z =⇩E x] = [λz z =⇩E y]›
6681 moreover AOT_have ‹[λz z =⇩E x]x›
6682 apply (rule "β←C"(1))
6683 apply "cqt:2[lambda]"
6684 apply (fact "cqt:2[const_var]"[axiom_inst])
6685 using "ord=Eequiv:1"[THEN "→E", OF 0[THEN "&E"(1)]].
6686 ultimately AOT_have ‹[λz z =⇩E y]x› using "rule=E" by fast
6687 AOT_hence ‹x =⇩E y› using "β→C"(1) by blast
6688 AOT_hence ‹x = y› by (metis "=E-simple:2" "vdash-properties:6")
6689 AOT_thus ‹x = y & ¬(x = y)› using 1 "&I" by blast
6690next
6691 AOT_assume ‹[λz z =⇩E x] ≠ [λz z =⇩E y]›
6692 AOT_hence 0: ‹¬([λz z =⇩E x] = [λz z =⇩E y])› using "≡⇩d⇩fE"[OF "=-infix"] by blast
6693 AOT_have ‹[λz z =⇩E x]↓› by "cqt:2[lambda]"
6694 AOT_hence ‹[λz z =⇩E x] = [λz z =⇩E x]›
6695 by (metis "rule=I:1")
6696 moreover AOT_assume ‹x = y›
6697 ultimately AOT_have ‹[λz z =⇩E x] = [λz z =⇩E y]›
6698 using "rule=E" by fast
6699 AOT_thus ‹[λz z =⇩E x] = [λz z =⇩E y] & ¬([λz z =⇩E x] = [λz z =⇩E y])›
6700 using 0 "&I" by blast
6701qed
6702
6703AOT_theorem "ord=E2:2": ‹(O!x & O!y) → (x ≠ y ≡ [λz z = x] ≠ [λz z = y])›
6704proof (rule "→I"; rule "≡I"; rule "→I"; rule "≡⇩d⇩fI"[OF "=-infix"]; rule "raa-cor:2")
6705 AOT_assume 0: ‹O!x & O!y›
6706 AOT_assume ‹x ≠ y›
6707 AOT_hence 1: ‹¬(x = y)› using "≡⇩d⇩fE"[OF "=-infix"] by blast
6708 AOT_assume ‹[λz z = x] = [λz z = y]›
6709 moreover AOT_have ‹[λz z = x]x›
6710 apply (rule "β←C"(1))
6711 apply (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
6712 apply (fact "cqt:2[const_var]"[axiom_inst])
6713 by (simp add: "id-eq:1")
6714 ultimately AOT_have ‹[λz z = y]x› using "rule=E" by fast
6715 AOT_hence ‹x = y› using "β→C"(1) by blast
6716 AOT_thus ‹x = y & ¬(x = y)› using 1 "&I" by blast
6717next
6718 AOT_assume 0: ‹O!x & O!y›
6719 AOT_assume ‹[λz z = x] ≠ [λz z = y]›
6720 AOT_hence 1: ‹¬([λz z = x] = [λz z = y])› using "≡⇩d⇩fE"[OF "=-infix"] by blast
6721 AOT_have ‹[λz z = x]↓› by (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
6722 AOT_hence ‹[λz z = x] = [λz z = x]›
6723 by (metis "rule=I:1")
6724 moreover AOT_assume ‹x = y›
6725 ultimately AOT_have ‹[λz z = x] = [λz z = y]›
6726 using "rule=E" by fast
6727 AOT_thus ‹[λz z = x] = [λz z = y] & ¬([λz z = x] = [λz z = y])›
6728 using 1 "&I" by blast
6729qed
6730
6731AOT_theorem ordnecfail: ‹O!x → □¬∃F x[F]›
6732 by (meson "RM:1" "deduction-theorem" nocoder "oa-facts:1" "vdash-properties:10" "vdash-properties:1[2]")
6733
6734AOT_theorem "ab-obey:1": ‹(A!x & A!y) → (∀F (x[F] ≡ y[F]) → x = y)›
6735proof (rule "→I"; rule "→I")
6736 AOT_assume 1: ‹A!x & A!y›
6737 AOT_assume ‹∀F (x[F] ≡ y[F])›
6738 AOT_hence ‹x[F] ≡ y[F]› for F using "∀E" by blast
6739 AOT_hence ‹□(x[F] ≡ y[F])› for F by (metis "en-eq:6[1]" "≡E"(1))
6740 AOT_hence ‹∀F □(x[F] ≡ y[F])› by (rule GEN)
6741 AOT_hence ‹□∀F (x[F] ≡ y[F])› by (rule BF[THEN "→E"])
6742 AOT_thus ‹x = y›
6743 using "≡⇩d⇩fI"[OF "identity:1", OF "∨I"(2)] 1 "&I" by blast
6744qed
6745
6746AOT_theorem "ab-obey:2": ‹(∃F (x[F] & ¬y[F]) ∨ ∃F (y[F] & ¬x[F])) → x ≠ y›
6747proof (rule "→I"; rule "≡⇩d⇩fI"[OF "=-infix"]; rule "raa-cor:2")
6748 AOT_assume 1: ‹x = y›
6749 AOT_assume ‹∃F (x[F] & ¬y[F]) ∨ ∃F (y[F] & ¬x[F])›
6750 moreover {
6751 AOT_assume ‹∃F (x[F] & ¬y[F])›
6752 then AOT_obtain F where ‹x[F] & ¬y[F]› using "∃E"[rotated] by blast
6753 moreover AOT_have ‹y[F]› using calculation[THEN "&E"(1)] 1 "rule=E" by fast
6754 ultimately AOT_have ‹p & ¬p› for p by (metis "Conjunction Simplification"(2) "modus-tollens:2" "raa-cor:3")
6755 }
6756 moreover {
6757 AOT_assume ‹∃F (y[F] & ¬x[F])›
6758 then AOT_obtain F where ‹y[F] & ¬x[F]› using "∃E"[rotated] by blast
6759 moreover AOT_have ‹¬y[F]› using calculation[THEN "&E"(2)] 1 "rule=E" by fast
6760 ultimately AOT_have ‹p & ¬p› for p by (metis "Conjunction Simplification"(1) "modus-tollens:1" "raa-cor:3")
6761 }
6762 ultimately AOT_show ‹p & ¬p› for p by (metis "∨E"(3) "raa-cor:1")
6763qed
6764
6765AOT_theorem "encoders-are-abstract": ‹∃F x[F] → A!x›
6766 by (meson "deduction-theorem" "≡E"(2) "modus-tollens:2" nocoder
6767 "oa-contingent:3" "vdash-properties:1[2]")
6768
6769AOT_theorem "denote=:1": ‹∀H∃x x[H]›
6770 by (rule GEN; rule "existence:2[1]"[THEN "≡⇩d⇩fE"]; fact "cqt:2[const_var]"[axiom_inst])
6771
6772AOT_theorem "denote=:2": ‹∀G∃x⇩1...∃x⇩n x⇩1...x⇩n[H]›
6773 by (rule GEN; rule "existence:2"[THEN "≡⇩d⇩fE"]; fact "cqt:2[const_var]"[axiom_inst])
6774
6775AOT_theorem "denote=:2[2]": ‹∀G∃x⇩1∃x⇩2 x⇩1x⇩2[H]›
6776 by (rule GEN; rule "existence:2[2]"[THEN "≡⇩d⇩fE"]; fact "cqt:2[const_var]"[axiom_inst])
6777
6778AOT_theorem "denote=:2[3]": ‹∀G∃x⇩1∃x⇩2∃x⇩3 x⇩1x⇩2x⇩3[H]›
6779 by (rule GEN; rule "existence:2[3]"[THEN "≡⇩d⇩fE"]; fact "cqt:2[const_var]"[axiom_inst])
6780
6781AOT_theorem "denote=:2[4]": ‹∀G∃x⇩1∃x⇩2∃x⇩3∃x⇩4 x⇩1x⇩2x⇩3x⇩4[H]›
6782 by (rule GEN; rule "existence:2[4]"[THEN "≡⇩d⇩fE"]; fact "cqt:2[const_var]"[axiom_inst])
6783
6784AOT_theorem "denote=:3": ‹∃x x[Π] ≡ ∃H (H = Π)›
6785 using "existence:2[1]" "free-thms:1" "≡E"(2) "≡E"(5) "Commutativity of ≡" "≡Df" by blast
6786
6787AOT_theorem "denote=:4": ‹(∃x⇩1...∃x⇩n x⇩1...x⇩n[Π]) ≡ ∃H (H = Π)›
6788 using "existence:2" "free-thms:1" "≡E"(6) "≡Df" by blast
6789
6790AOT_theorem "denote=:4[2]": ‹(∃x⇩1∃x⇩2 x⇩1x⇩2[Π]) ≡ ∃H (H = Π)›
6791 using "existence:2[2]" "free-thms:1" "≡E"(6) "≡Df" by blast
6792
6793AOT_theorem "denote=:4[3]": ‹(∃x⇩1∃x⇩2∃x⇩3 x⇩1x⇩2x⇩3[Π]) ≡ ∃H (H = Π)›
6794 using "existence:2[3]" "free-thms:1" "≡E"(6) "≡Df" by blast
6795
6796AOT_theorem "denote=:4[4]": ‹(∃x⇩1∃x⇩2∃x⇩3∃x⇩4 x⇩1x⇩2x⇩3x⇩4[Π]) ≡ ∃H (H = Π)›
6797 using "existence:2[4]" "free-thms:1" "≡E"(6) "≡Df" by blast
6798
6799AOT_theorem "A-objects!": ‹∃!x (A!x & ∀F (x[F] ≡ φ{F}))›
6800proof (rule "uniqueness:1"[THEN "≡⇩d⇩fI"])
6801 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ φ{F})›
6802 using "A-objects"[axiom_inst] "∃E"[rotated] by blast
6803 AOT_have ‹(A!β & ∀F (β[F] ≡ φ{F})) → β = a› for β
6804 proof (rule "→I")
6805 AOT_assume β_prop: ‹[A!]β & ∀F (β[F] ≡ φ{F})›
6806 AOT_hence ‹β[F] ≡ φ{F}› for F using "∀E" "&E" by blast
6807 AOT_hence ‹β[F] ≡ a[F]› for F
6808 using a_prop[THEN "&E"(2)] "∀E" "≡E"(2) "≡E"(5) "Commutativity of ≡" by fast
6809 AOT_hence ‹∀F (β[F] ≡ a[F])› by (rule GEN)
6810 AOT_thus ‹β = a›
6811 using "ab-obey:1"[THEN "→E", OF "&I"[OF β_prop[THEN "&E"(1)], OF a_prop[THEN "&E"(1)]], THEN "→E"] by blast
6812 qed
6813 AOT_hence ‹∀β ((A!β & ∀F (β[F] ≡ φ{F})) → β = a)› by (rule GEN)
6814 AOT_thus ‹∃α ([A!]α & ∀F (α[F] ≡ φ{F}) & ∀β ([A!]β & ∀F (β[F] ≡ φ{F}) → β = α)) ›
6815 using "∃I" using a_prop "&I" by fast
6816qed
6817
6818AOT_theorem "obj-oth:1": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y))›
6819 using "A-objects!" by fast
6820
6821AOT_theorem "obj-oth:2": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y & [F]z))›
6822 using "A-objects!" by fast
6823
6824AOT_theorem "obj-oth:3": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y ∨ [F]z))›
6825 using "A-objects!" by fast
6826
6827AOT_theorem "obj-oth:4": ‹∃!x (A!x & ∀F (x[F] ≡ □[F]y))›
6828 using "A-objects!" by fast
6829
6830AOT_theorem "obj-oth:5": ‹∃!x (A!x & ∀F (x[F] ≡ F = G))›
6831 using "A-objects!" by fast
6832
6833AOT_theorem "obj-oth:6": ‹∃!x (A!x & ∀F (x[F] ≡ □∀y([G]y → [F]y)))›
6834 using "A-objects!" by fast
6835
6836AOT_theorem "A-descriptions": ‹❙ιx (A!x & ∀F (x[F] ≡ φ{F}))↓›
6837 by (rule "A-Exists:2"[THEN "≡E"(2)]; rule "RA[2]"; rule "A-objects!")
6838
6839AOT_act_theorem "thm-can-terms2": ‹y = ❙ιx(A!x & ∀F (x[F] ≡ φ{F})) → (A!y & ∀F (y[F] ≡ φ{F}))›
6840 using "y-in:2" by blast
6841
6842AOT_theorem "can-ab2": ‹y = ❙ιx(A!x & ∀F (x[F] ≡ φ{F})) → A!y›
6843proof(rule "→I")
6844 AOT_assume ‹y = ❙ιx(A!x & ∀F (x[F] ≡ φ{F}))›
6845 AOT_hence ‹❙𝒜(A!y & ∀F (y[F] ≡ φ{F}))›
6846 using "actual-desc:2"[THEN "→E"] by blast
6847 AOT_hence ‹❙𝒜A!y› by (metis "Act-Basic:2" "&E"(1) "≡E"(1))
6848 AOT_thus ‹A!y› by (metis "≡E"(2) "oa-facts:8")
6849qed
6850
6851AOT_act_theorem "desc-encode:1": ‹❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}›
6852proof -
6853 AOT_have ‹❙ιx(A!x & ∀F (x[F] ≡ φ{F}))↓›
6854 by (simp add: "A-descriptions")
6855 AOT_hence ‹A!❙ιx(A!x & ∀F (x[F] ≡ φ{F})) & ∀F (❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6856 using "y-in:3"[THEN "→E"] by blast
6857 AOT_thus ‹❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}›
6858 using "&E" "∀E" by blast
6859qed
6860
6861AOT_act_theorem "desc-encode:2": ‹❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ φ{G}›
6862 using "desc-encode:1".
6863
6864AOT_theorem "desc-nec-encode:1": ‹❙ιx (A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ ❙𝒜φ{F}›
6865proof -
6866 AOT_have 0: ‹❙ιx(A!x & ∀F (x[F] ≡ φ{F}))↓›
6867 by (simp add: "A-descriptions")
6868 AOT_hence ‹❙𝒜(A!❙ιx(A!x & ∀F (x[F] ≡ φ{F})) & ∀F (❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}))›
6869 using "actual-desc:4"[THEN "→E"] by blast
6870 AOT_hence ‹❙𝒜∀F (❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6871 using "Act-Basic:2" "&E"(2) "≡E"(1) by blast
6872 AOT_hence ‹∀F ❙𝒜(❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6873 using "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]" by blast
6874 AOT_hence ‹❙𝒜(❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
6875 using "∀E" by blast
6876 AOT_hence ‹❙𝒜❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ ❙𝒜φ{F}›
6877 using "Act-Basic:5" "≡E"(1) by blast
6878 AOT_thus ‹❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ ❙𝒜φ{F}›
6879 using "en-eq:10[1]"[unvarify x⇩1, OF 0] "≡E"(6) by blast
6880qed
6881
6882AOT_theorem "desc-nec-encode:2": ‹❙ιx (A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ ❙𝒜φ{G}›
6883 using "desc-nec-encode:1".
6884
6885AOT_theorem "Box-desc-encode:1": ‹□φ{G} → ❙ιx(A!x & ∀F (x[F] ≡ φ{G}))[G]›
6886 by (rule "→I"; rule "desc-nec-encode:2"[THEN "≡E"(2)])
6887 (meson "nec-imp-act" "vdash-properties:10")
6888
6889AOT_theorem "Box-desc-encode:2": ‹□φ{G} → □(❙ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})›
6890proof(rule CP)
6891 AOT_assume ‹□φ{G}›
6892 AOT_hence ‹□□φ{G}› by (metis "S5Basic:6" "≡E"(1))
6893 moreover AOT_have ‹□□φ{G} → □(❙ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})›
6894 proof (rule RM; rule "→I")
6895 AOT_modally_strict {
6896 AOT_assume 1: ‹□φ{G}›
6897 AOT_hence ‹❙ιx(A!x & ∀F (x[F] ≡ φ{G}))[G]› using "Box-desc-encode:1" "→E" by blast
6898 moreover AOT_have ‹φ{G}› using 1 by (meson "qml:2" "vdash-properties:10" "vdash-properties:1[2]")
6899 ultimately AOT_show ‹❙ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G}›
6900 using "deduction-theorem" "≡I" by simp
6901 }
6902 qed
6903 ultimately AOT_show ‹□(❙ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})› using "→E" by blast
6904qed
6905
6906definition rigid_condition where ‹rigid_condition φ ≡ ∀v . [v ⊨ ∀α (φ{α} → □φ{α})]›
6907syntax rigid_condition :: ‹id_position ⇒ AOT_prop› ("RIGID'_CONDITION'(_')")
6908
6909AOT_theorem "strict-can:1[E]": assumes ‹RIGID_CONDITION(φ)›
6910 shows ‹∀α (φ{α} → □φ{α})›
6911 using assms[unfolded rigid_condition_def] by auto
6912
6913AOT_theorem "strict-can:1[I]":
6914 assumes ‹❙⊢⇩□ ∀α (φ{α} → □φ{α})›
6915 shows ‹RIGID_CONDITION(φ)›
6916 using assms rigid_condition_def by auto
6917
6918AOT_theorem "box-phi-a:1": assumes ‹RIGID_CONDITION(φ)›
6919 shows ‹(A!x & ∀F (x[F] ≡ φ{F})) → □(A!x & ∀F (x[F] ≡ φ{F}))›
6920proof (rule "→I")
6921 AOT_assume a: ‹A!x & ∀F (x[F] ≡ φ{F})›
6922 AOT_hence b: ‹□A!x› by (metis "Conjunction Simplification"(1) "oa-facts:2" "vdash-properties:10")
6923 AOT_have ‹x[F] ≡ φ{F}› for F using a[THEN "&E"(2)] "∀E" by blast
6924 moreover AOT_have ‹□(x[F] → □x[F])› for F by (meson "pre-en-eq:1[1]" RN)
6925 moreover AOT_have ‹□(φ{F} → □φ{F})› for F using RN "strict-can:1[E]"[OF assms] "∀E" by blast
6926 ultimately AOT_have ‹□(x[F] ≡ φ{F})› for F
6927 using "sc-eq-box-box:5" "qml:2"[axiom_inst, THEN "→E"] "→E" "&I" by metis
6928 AOT_hence ‹∀F □(x[F] ≡ φ{F})› by (rule GEN)
6929 AOT_hence ‹□∀F (x[F] ≡ φ{F})› by (rule BF[THEN "→E"])
6930 AOT_thus ‹□([A!]x & ∀F (x[F] ≡ φ{F}))›
6931 using b "KBasic:3" "≡S"(1) "≡E"(2) by blast
6932qed
6933
6934AOT_theorem "box-phi-a:2": assumes ‹RIGID_CONDITION(φ)›
6935 shows ‹y = ❙ιx(A!x & ∀F (x[F] ≡ φ{F})) → (A!y & ∀F (y[F] ≡ φ{F}))›
6936proof(rule "→I")
6937 AOT_assume ‹y = ❙ιx(A!x & ∀F (x[F] ≡ φ{F}))›
6938 AOT_hence ‹❙𝒜(A!y & ∀F (y[F] ≡ φ{F}))› using "actual-desc:2"[THEN "→E"] by fast
6939 AOT_hence abs: ‹❙𝒜A!y› and ‹❙𝒜∀F (y[F] ≡ φ{F})›
6940 using "Act-Basic:2" "&E" "≡E"(1) by blast+
6941 AOT_hence ‹∀F ❙𝒜(y[F] ≡ φ{F})› by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
6942 AOT_hence ‹❙𝒜(y[F] ≡ φ{F})› for F using "∀E" by blast
6943 AOT_hence ‹❙𝒜y[F] ≡ ❙𝒜φ{F}› for F by (metis "Act-Basic:5" "≡E"(1))
6944 AOT_hence ‹y[F] ≡ φ{F}› for F
6945 using "sc-eq-fur:2"[THEN "→E", OF "strict-can:1[E]"[OF assms, THEN "∀E"(2)[where β=F], THEN RN]]
6946 by (metis "en-eq:10[1]" "≡E"(6))
6947 AOT_hence ‹∀F (y[F] ≡ φ{F})› by (rule GEN)
6948 AOT_thus ‹[A!]y & ∀F (y[F] ≡ φ{F})› using abs "&I" "≡E"(2) "oa-facts:8" by blast
6949qed
6950
6951AOT_theorem "box-phi-a:3": assumes ‹RIGID_CONDITION(φ)›
6952 shows ‹❙ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}›
6953 using "desc-nec-encode:2"
6954 "sc-eq-fur:2"[THEN "→E", OF "strict-can:1[E]"[OF assms, THEN "∀E"(2)[where β=F], THEN RN]]
6955 "≡E"(5) by blast
6956
6957AOT_define Null :: ‹τ ⇒ φ› ("Null'(_')")
6958 "df-null-uni:1": ‹Null(x) ≡⇩d⇩f A!x & ¬∃F x[F]›
6959
6960AOT_define Universal :: ‹τ ⇒ φ› ("Universal'(_')")
6961 "df-null-uni:2": ‹Universal(x) ≡⇩d⇩f A!x & ∀F x[F]›
6962
6963AOT_theorem "null-uni-uniq:1": ‹∃!x Null(x)›
6964proof (rule "uniqueness:1"[THEN "≡⇩d⇩fI"])
6965 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ¬(F = F))›
6966 using "A-objects"[axiom_inst] "∃E"[rotated] by fast
6967 AOT_have a_null: ‹¬a[F]› for F
6968 proof (rule "raa-cor:2")
6969 AOT_assume ‹a[F]›
6970 AOT_hence ‹¬(F = F)› using a_prop[THEN "&E"(2)] "∀E" "≡E" by blast
6971 AOT_hence ‹F = F & ¬(F = F)› by (metis "id-eq:1" "raa-cor:3")
6972 AOT_thus ‹p & ¬p› for p by (metis "raa-cor:1")
6973 qed
6974 AOT_have ‹Null(a) & ∀β (Null(β) → β = a)›
6975 proof (rule "&I")
6976 AOT_have ‹¬∃F a[F]› using a_null by (metis "instantiation" "reductio-aa:1")
6977 AOT_thus ‹Null(a)›
6978 using "df-null-uni:1"[THEN "≡⇩d⇩fI"] a_prop[THEN "&E"(1)] "&I" by metis
6979 next
6980 AOT_show ‹∀β (Null(β) → β = a)›
6981 proof (rule GEN; rule "→I")
6982 fix β
6983 AOT_assume a: ‹Null(β)›
6984 AOT_hence ‹¬∃F β[F]›
6985 using "df-null-uni:1"[THEN "≡⇩d⇩fE"] "&E" by blast
6986 AOT_hence β_null: ‹¬β[F]› for F by (metis "existential:2[const_var]" "reductio-aa:1")
6987 AOT_have ‹∀F (β[F] ≡ a[F])›
6988 apply (rule GEN; rule "≡I"; rule CP)
6989 using "raa-cor:3" β_null a_null by blast+
6990 moreover AOT_have ‹A!β› using a "df-null-uni:1"[THEN "≡⇩d⇩fE"] "&E" by blast
6991 ultimately AOT_show ‹β = a›
6992 using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"] "&I" by blast
6993 qed
6994 qed
6995 AOT_thus ‹∃α (Null(α) & ∀β (Null(β) → β = α))› using "∃I"(2) by fast
6996qed
6997
6998AOT_theorem "null-uni-uniq:2": ‹∃!x Universal(x)›
6999proof (rule "uniqueness:1"[THEN "≡⇩d⇩fI"])
7000 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ F = F)›
7001 using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7002 AOT_hence aF: ‹a[F]› for F using "&E" "∀E" "≡E" "id-eq:1" by fast
7003 AOT_hence ‹Universal(a)›
7004 using "df-null-uni:2"[THEN "≡⇩d⇩fI"] "&I" a_prop[THEN "&E"(1)] GEN by blast
7005 moreover AOT_have ‹∀β (Universal(β) → β = a)›
7006 proof (rule GEN; rule "→I")
7007 fix β
7008 AOT_assume ‹Universal(β)›
7009 AOT_hence abs_β: ‹A!β› and ‹β[F]› for F using "df-null-uni:2"[THEN "≡⇩d⇩fE"] "&E" "∀E" by blast+
7010 AOT_hence ‹β[F] ≡ a[F]› for F using aF by (metis "deduction-theorem" "≡I")
7011 AOT_hence ‹∀F (β[F] ≡ a[F])› by (rule GEN)
7012 AOT_thus ‹β = a›
7013 using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"] "&I" abs_β by blast
7014 qed
7015 ultimately AOT_show ‹∃α (Universal(α) & ∀β (Universal(β) → β = α))›
7016 using "&I" "∃I" by fast
7017qed
7018
7019AOT_theorem "null-uni-uniq:3": ‹❙ιx Null(x)↓›
7020 using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:1" by blast
7021
7022AOT_theorem "null-uni-uniq:4": ‹❙ιx Universal(x)↓›
7023 using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:2" by blast
7024
7025AOT_define Null_object :: ‹κ⇩s› (‹a⇩∅›)
7026 "df-null-uni-terms:1": ‹a⇩∅ =⇩d⇩f ❙ιx Null(x)›
7027
7028AOT_define Universal_object :: ‹κ⇩s› (‹a⇩V›)
7029 "df-null-uni-terms:2": ‹a⇩V =⇩d⇩f ❙ιx Universal(x)›
7030
7031AOT_theorem "null-uni-facts:1": ‹Null(x) → □Null(x)›
7032proof (rule "→I")
7033 AOT_assume ‹Null(x)›
7034 AOT_hence x_abs: ‹A!x› and x_null: ‹¬∃F x[F]›
7035 using "df-null-uni:1"[THEN "≡⇩d⇩fE"] "&E" by blast+
7036 AOT_have ‹¬x[F]› for F using x_null
7037 using "existential:2[const_var]" "reductio-aa:1"
7038 by metis
7039 AOT_hence ‹□¬x[F]› for F by (metis "en-eq:7[1]" "≡E"(1))
7040 AOT_hence ‹∀F □¬x[F]› by (rule GEN)
7041 AOT_hence ‹□∀F ¬x[F]› by (rule BF[THEN "→E"])
7042 moreover AOT_have ‹□∀F ¬x[F] → □¬∃F x[F]›
7043 apply (rule RM)
7044 by (metis (full_types) "instantiation" "cqt:2[const_var]" "deduction-theorem"
7045 "reductio-aa:1" "rule-ui:1" "vdash-properties:1[2]")
7046 ultimately AOT_have ‹□¬∃F x[F]›
7047 by (metis "→E")
7048 moreover AOT_have ‹□A!x› using x_abs
7049 using "oa-facts:2" "vdash-properties:10" by blast
7050 ultimately AOT_have r: ‹□(A!x & ¬∃F x[F])›
7051 by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
7052 AOT_show ‹□Null(x)›
7053 by (AOT_subst ‹Null(x)› ‹A!x & ¬∃F x[F]›)
7054 (auto simp: "df-null-uni:1" "≡Df" r)
7055qed
7056
7057AOT_theorem "null-uni-facts:2": ‹Universal(x) → □Universal(x)›
7058proof (rule "→I")
7059 AOT_assume ‹Universal(x)›
7060 AOT_hence x_abs: ‹A!x› and x_univ: ‹∀F x[F]›
7061 using "df-null-uni:2"[THEN "≡⇩d⇩fE"] "&E" by blast+
7062 AOT_have ‹x[F]› for F using x_univ "∀E" by blast
7063 AOT_hence ‹□x[F]› for F by (metis "en-eq:2[1]" "≡E"(1))
7064 AOT_hence ‹∀F □x[F]› by (rule GEN)
7065 AOT_hence ‹□∀F x[F]› by (rule BF[THEN "→E"])
7066 moreover AOT_have ‹□A!x› using x_abs
7067 using "oa-facts:2" "vdash-properties:10" by blast
7068 ultimately AOT_have r: ‹□(A!x & ∀F x[F])›
7069 by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
7070 AOT_show ‹□Universal(x)›
7071 by (AOT_subst ‹Universal(x)› ‹A!x & ∀F x[F]›)
7072 (auto simp add: "df-null-uni:2" "≡Df" r)
7073qed
7074
7075AOT_theorem "null-uni-facts:3": ‹Null(a⇩∅)›
7076 apply (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:1"])
7077 apply (simp add: "null-uni-uniq:3")
7078 using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:3"]
7079 "sc-eq-fur:2"[THEN "→E", OF "null-uni-facts:1"[unvarify x, THEN RN, OF "null-uni-uniq:3"], THEN "≡E"(1)]
7080 by blast
7081
7082AOT_theorem "null-uni-facts:4": ‹Universal(a⇩V)›
7083 apply (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:2"])
7084 apply (simp add: "null-uni-uniq:4")
7085 using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:4"]
7086 "sc-eq-fur:2"[THEN "→E", OF "null-uni-facts:2"[unvarify x, THEN RN, OF "null-uni-uniq:4"], THEN "≡E"(1)]
7087 by blast
7088
7089AOT_theorem "null-uni-facts:5": ‹a⇩∅ ≠ a⇩V›
7090proof (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"];
7091 rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"];
7092 rule "≡⇩d⇩fI"[OF "=-infix"];
7093 rule "raa-cor:2")
7094 AOT_obtain x where nullx: ‹Null(x)›
7095 by (metis "instantiation" "df-null-uni-terms:1" "existential:1" "null-uni-facts:3"
7096 "null-uni-uniq:3" "rule-id-df:2:b[zero]")
7097 AOT_hence act_null: ‹❙𝒜Null(x)› by (metis "nec-imp-act" "null-uni-facts:1" "vdash-properties:10")
7098 AOT_assume ‹❙ιx Null(x) = ❙ιx Universal(x)›
7099 AOT_hence ‹❙𝒜∀x(Null(x) ≡ Universal(x))›
7100 using "actual-desc:5"[THEN "→E"] by blast
7101 AOT_hence ‹∀x ❙𝒜(Null(x) ≡ Universal(x))›
7102 by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
7103 AOT_hence ‹❙𝒜Null(x) ≡ ❙𝒜Universal(x)›
7104 using "Act-Basic:5" "≡E"(1) "rule-ui:3" by blast
7105 AOT_hence ‹❙𝒜Universal(x)› using act_null "≡E" by blast
7106 AOT_hence ‹Universal(x)› by (metis RN "≡E"(1) "null-uni-facts:2" "sc-eq-fur:2" "vdash-properties:10")
7107 AOT_hence ‹∀F x[F]› using "≡⇩d⇩fE"[OF "df-null-uni:2"] "&E" by metis
7108 moreover AOT_have ‹¬∃F x[F]› using nullx "≡⇩d⇩fE"[OF "df-null-uni:1"] "&E" by metis
7109 ultimately AOT_show ‹p & ¬p› for p by (metis "cqt-further:1" "raa-cor:3" "vdash-properties:10")
7110qed
7111
7112AOT_theorem "null-uni-facts:6": ‹a⇩∅ = ❙ιx(A!x & ∀F (x[F] ≡ F ≠ F))›
7113proof (rule "ab-obey:1"[unvarify x y, THEN "→E", THEN "→E"])
7114 AOT_show ‹❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))↓›
7115 by (simp add: "A-descriptions")
7116next
7117 AOT_show ‹a⇩∅↓›
7118 by (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
7119 (simp add: "null-uni-uniq:3")
7120next
7121 AOT_have ‹❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))↓›
7122 by (simp add: "A-descriptions")
7123 AOT_hence 1: ‹❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F)) = ❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))›
7124 using "rule=I:1" by blast
7125 AOT_show ‹[A!]a⇩∅ & [A!]❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))›
7126 apply (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"]; rule "&I")
7127 apply (meson "≡⇩d⇩fE" "Conjunction Simplification"(1) "df-null-uni:1" "df-null-uni-terms:1" "null-uni-facts:3" "null-uni-uniq:3" "rule-id-df:2:a[zero]" "vdash-properties:10")
7128 using "can-ab2"[unvarify y, OF "A-descriptions", THEN "→E", OF 1].
7129next
7130 AOT_show ‹∀F (a⇩∅[F] ≡ ❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F])›
7131 proof (rule GEN)
7132 fix F
7133 AOT_have ‹¬a⇩∅[F]›
7134 by (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
7135 (metis (no_types, lifting) "≡⇩d⇩fE" "&E"(2) "∨I"(2) "∨E"(3)
7136 "df-null-uni:1" "df-null-uni-terms:1" "existential:2[const_var]" "null-uni-facts:3"
7137 "raa-cor:2" "rule-id-df:2:a[zero]" "russell-axiom[enc,1].ψ_denotes_asm")
7138 moreover AOT_have ‹¬❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7139 proof(rule "raa-cor:2")
7140 AOT_assume 0: ‹❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7141 AOT_hence ‹❙𝒜(F ≠ F)› using "desc-nec-encode:2"[THEN "≡E"(1), OF 0] by blast
7142 moreover AOT_have ‹¬❙𝒜(F ≠ F)›
7143 using "≡⇩d⇩fE" "id-act:2" "id-eq:1" "≡E"(2) "=-infix" "raa-cor:3" by blast
7144 ultimately AOT_show ‹❙𝒜(F ≠ F) & ¬❙𝒜(F ≠ F)› by (rule "&I")
7145 qed
7146 ultimately AOT_show ‹a⇩∅[F] ≡ ❙ιx([A!]x & ∀F (x[F] ≡ F ≠ F))[F]›
7147 using "deduction-theorem" "≡I" "raa-cor:4" by blast
7148 qed
7149qed
7150
7151AOT_theorem "null-uni-facts:7": ‹a⇩V = ❙ιx(A!x & ∀F (x[F] ≡ F = F))›
7152proof (rule "ab-obey:1"[unvarify x y, THEN "→E", THEN "→E"])
7153 AOT_show ‹❙ιx([A!]x & ∀F (x[F] ≡ F = F))↓›
7154 by (simp add: "A-descriptions")
7155next
7156 AOT_show ‹a⇩V↓›
7157 by (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
7158 (simp add: "null-uni-uniq:4")
7159next
7160 AOT_have ‹❙ιx([A!]x & ∀F (x[F] ≡ F = F))↓›
7161 by (simp add: "A-descriptions")
7162 AOT_hence 1: ‹❙ιx([A!]x & ∀F (x[F] ≡ F = F)) = ❙ιx([A!]x & ∀F (x[F] ≡ F = F))›
7163 using "rule=I:1" by blast
7164 AOT_show ‹[A!]a⇩V & [A!]❙ιx([A!]x & ∀F (x[F] ≡ F = F))›
7165 apply (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"]; rule "&I")
7166 apply (meson "≡⇩d⇩fE" "Conjunction Simplification"(1) "df-null-uni:2" "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4" "rule-id-df:2:a[zero]" "vdash-properties:10")
7167 using "can-ab2"[unvarify y, OF "A-descriptions", THEN "→E", OF 1].
7168next
7169 AOT_show ‹∀F (a⇩V[F] ≡ ❙ιx([A!]x & ∀F (x[F] ≡ F = F))[F])›
7170 proof (rule GEN)
7171 fix F
7172 AOT_have ‹a⇩V[F]›
7173 apply (rule "=⇩d⇩fI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
7174 using "≡⇩d⇩fE" "&E"(2) "df-null-uni:2" "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4" "rule-id-df:2:a[zero]" "rule-ui:3" by blast
7175 moreover AOT_have ‹❙ιx([A!]x & ∀F (x[F] ≡ F = F))[F]›
7176 using "RA[2]" "desc-nec-encode:2" "id-eq:1" "≡E"(2) by fastforce
7177 ultimately AOT_show ‹a⇩V[F] ≡ ❙ιx([A!]x & ∀F (x[F] ≡ F = F))[F]›
7178 using "deduction-theorem" "≡I" by simp
7179 qed
7180qed
7181
7182AOT_theorem "aclassical:1": ‹∀R∃x∃y(A!x & A!y & x ≠ y & [λz [R]zx] = [λz [R]zy])›
7183proof(rule GEN)
7184 fix R
7185 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]zy] & ¬y[F]))›
7186 using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7187 AOT_have a_enc: ‹a[λz [R]za]›
7188 proof (rule "raa-cor:1")
7189 AOT_assume 0: ‹¬a[λz [R]za]›
7190 AOT_hence ‹¬∃y(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7191 by (rule a_prop[THEN "&E"(2), THEN "∀E"(1)[where τ="«[λz [R]za]»"],
7192 THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7193 "cqt:2[lambda]"
7194 AOT_hence ‹∀y ¬(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7195 using "cqt-further:4" "vdash-properties:10" by blast
7196 AOT_hence ‹¬(A!a & [λz [R]za] = [λz [R]za] & ¬a[λz [R]za])› using "∀E" by blast
7197 AOT_hence ‹(A!a & [λz [R]za] = [λz [R]za]) → a[λz [R]za]›
7198 by (metis "&I" "deduction-theorem" "raa-cor:3")
7199 moreover AOT_have ‹[λz [R]za] = [λz [R]za]›
7200 by (rule "=I") "cqt:2[lambda]"
7201 ultimately AOT_have ‹a[λz [R]za]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7202 AOT_thus ‹a[λz [R]za] & ¬a[λz [R]za]›
7203 using 0 "&I" by blast
7204 qed
7205 AOT_hence ‹∃y(A!y & [λz [R]za] = [λz [R]zy] & ¬y[λz [R]za])›
7206 by (rule a_prop[THEN "&E"(2), THEN "∀E"(1), THEN "≡E"(1), rotated]) "cqt:2[lambda]"
7207 then AOT_obtain b where b_prop: ‹A!b & [λz [R]za] = [λz [R]zb] & ¬b[λz [R]za]›
7208 using "∃E"[rotated] by blast
7209 AOT_have ‹a ≠ b›
7210 apply (rule "≡⇩d⇩fI"[OF "=-infix"])
7211 using a_enc b_prop[THEN "&E"(2)]
7212 using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7213 AOT_hence ‹A!a & A!b & a ≠ b & [λz [R]za] = [λz [R]zb]›
7214 using b_prop "&E" a_prop "&I" by meson
7215 AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λz [R]za] = [λz [R]zy])› by (rule "∃I")
7216 AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λz [R]zx] = [λz [R]zy])› by (rule "∃I")
7217qed
7218
7219AOT_theorem "aclassical:2": ‹∀R∃x∃y(A!x & A!y & x ≠ y & [λz [R]xz] = [λz [R]yz])›
7220proof(rule GEN)
7221 fix R
7222 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]yz] & ¬y[F]))›
7223 using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7224 AOT_have a_enc: ‹a[λz [R]az]›
7225 proof (rule "raa-cor:1")
7226 AOT_assume 0: ‹¬a[λz [R]az]›
7227 AOT_hence ‹¬∃y(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7228 by (rule a_prop[THEN "&E"(2), THEN "∀E"(1)[where τ="«[λz [R]az]»"],
7229 THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7230 "cqt:2[lambda]"
7231 AOT_hence ‹∀y ¬(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7232 using "cqt-further:4" "vdash-properties:10" by blast
7233 AOT_hence ‹¬(A!a & [λz [R]az] = [λz [R]az] & ¬a[λz [R]az])› using "∀E" by blast
7234 AOT_hence ‹(A!a & [λz [R]az] = [λz [R]az]) → a[λz [R]az]›
7235 by (metis "&I" "deduction-theorem" "raa-cor:3")
7236 moreover AOT_have ‹[λz [R]az] = [λz [R]az]›
7237 by (rule "=I") "cqt:2[lambda]"
7238 ultimately AOT_have ‹a[λz [R]az]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7239 AOT_thus ‹a[λz [R]az] & ¬a[λz [R]az]›
7240 using 0 "&I" by blast
7241 qed
7242 AOT_hence ‹∃y(A!y & [λz [R]az] = [λz [R]yz] & ¬y[λz [R]az])›
7243 by (rule a_prop[THEN "&E"(2), THEN "∀E"(1), THEN "≡E"(1), rotated]) "cqt:2[lambda]"
7244 then AOT_obtain b where b_prop: ‹A!b & [λz [R]az] = [λz [R]bz] & ¬b[λz [R]az]›
7245 using "∃E"[rotated] by blast
7246 AOT_have ‹a ≠ b›
7247 apply (rule "≡⇩d⇩fI"[OF "=-infix"])
7248 using a_enc b_prop[THEN "&E"(2)]
7249 using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7250 AOT_hence ‹A!a & A!b & a ≠ b & [λz [R]az] = [λz [R]bz]›
7251 using b_prop "&E" a_prop "&I" by meson
7252 AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λz [R]az] = [λz [R]yz])› by (rule "∃I")
7253 AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λz [R]xz] = [λz [R]yz])› by (rule "∃I")
7254qed
7255
7256AOT_theorem "aclassical:3": ‹∀F∃x∃y(A!x & A!y & x ≠ y & [λ [F]x] = [λ [F]y])›
7257proof(rule GEN)
7258 fix R
7259 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ∃y(A!y & F = [λz [R]y] & ¬y[F]))›
7260 using "A-objects"[axiom_inst] "∃E"[rotated] by fast
7261 AOT_have ‹[λz [R]a]↓› by "cqt:2[lambda]"
7262
7263 then AOT_obtain S where S_def: ‹S = [λz [R]a]›
7264 by (metis "instantiation" "rule=I:1" "existential:1" id_sym)
7265 AOT_have a_enc: ‹a[S]›
7266 proof (rule "raa-cor:1")
7267 AOT_assume 0: ‹¬a[S]›
7268 AOT_hence ‹¬∃y(A!y & S = [λz [R]y] & ¬y[S])›
7269 by (rule a_prop[THEN "&E"(2), THEN "∀E"(2)[where β=S],
7270 THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1), rotated])
7271 AOT_hence ‹∀y ¬(A!y & S = [λz [R]y] & ¬y[S])›
7272 using "cqt-further:4" "vdash-properties:10" by blast
7273 AOT_hence ‹¬(A!a & S = [λz [R]a] & ¬a[S])› using "∀E" by blast
7274 AOT_hence ‹(A!a & S = [λz [R]a]) → a[S]›
7275 by (metis "&I" "deduction-theorem" "raa-cor:3")
7276 moreover AOT_have ‹S = [λz [R]a]› using S_def .
7277 ultimately AOT_have ‹a[S]› using a_prop[THEN "&E"(1)] "→E" "&I" by blast
7278 AOT_thus ‹a[λz [R]a] & ¬a[λz [R]a]› by (metis "0" "raa-cor:3")
7279 qed
7280 AOT_hence ‹∃y(A!y & S = [λz [R]y] & ¬y[S])›
7281 by (rule a_prop[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1), rotated])
7282 then AOT_obtain b where b_prop: ‹A!b & S = [λz [R]b] & ¬b[S]›
7283 using "∃E"[rotated] by blast
7284 AOT_have 1: ‹a ≠ b›
7285 apply (rule "≡⇩d⇩fI"[OF "=-infix"])
7286 using a_enc b_prop[THEN "&E"(2)]
7287 using "¬¬I" "rule=E" id_sym "≡E"(4) "oth-class-taut:3:a" "raa-cor:3" "reductio-aa:1" by fast
7288 AOT_have a: ‹[λ [R]a] = ([R]a)›
7289 apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
7290 by (meson "log-prop-prop:2")
7291 AOT_have b: ‹[λ [R]b] = ([R]b)›
7292 apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
7293 by (meson "log-prop-prop:2")
7294 AOT_have ‹[λ [R]a] = [λ [R]b]›
7295 apply (rule "rule=E"[rotated, OF a[THEN id_sym]])
7296 apply (rule "rule=E"[rotated, OF b[THEN id_sym]])
7297 apply (rule "identity:4"[THEN "≡⇩d⇩fI", OF "&I", rotated])
7298 apply (rule "rule=E"[rotated, OF S_def])
7299 using b_prop "&E" apply blast
7300 apply (safe intro!: "&I")
7301 by (simp add: "log-prop-prop:2")+
7302 AOT_hence ‹A!a & A!b & a ≠ b & [λ [R]a] = [λ [R]b]›
7303 using 1 a_prop[THEN "&E"(1)] b_prop[THEN "&E"(1), THEN "&E"(1)] "&I" by auto
7304 AOT_hence ‹∃y (A!a & A!y & a ≠ y & [λ [R]a] = [λ [R]y])› by (rule "∃I")
7305 AOT_thus ‹∃x∃y (A!x & A!y & x ≠ y & [λ [R]x] = [λ [R]y])› by (rule "∃I")
7306qed
7307
7308AOT_theorem aclassical2: ‹∃x∃y (A!x & A!y & x ≠ y & ∀F ([F]x ≡ [F]y))›
7309proof -
7310 AOT_have ‹∃x ∃y ([A!]x & [A!]y & x ≠ y &
7311 [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7312 by (rule "aclassical:1"[THEN "∀E"(1)[where τ="«[λxy ∀F ([F]x ≡ [F]y)]»"]])
7313 "cqt:2[lambda]"
7314 then AOT_obtain x where ‹∃y ([A!]x & [A!]y & x ≠ y &
7315 [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7316 using "∃E"[rotated] by blast
7317 then AOT_obtain y where 0: ‹([A!]x & [A!]y & x ≠ y &
7318 [λz [λxy ∀F ([F]x ≡ [F]y)]zx] = [λz [λxy ∀F ([F]x ≡ [F]y)]zy])›
7319 using "∃E"[rotated] by blast
7320 AOT_have ‹[λz [λxy ∀F ([F]x ≡ [F]y)]zx]x›
7321 apply (rule "β←C"(1))
7322 apply "cqt:2[lambda]"
7323 apply (fact "cqt:2[const_var]"[axiom_inst])
7324 apply (rule "β←C"(1))
7325 apply "cqt:2[lambda]"
7326 apply (simp add: "&I" "ex:1:a" prod_denotesI "rule-ui:3")
7327 by (simp add: "oth-class-taut:3:a" "universal-cor")
7328 AOT_hence ‹[λz [λxy ∀F ([F]x ≡ [F]y)]zy]x›
7329 by (rule "rule=E"[rotated, OF 0[THEN "&E"(2)]])
7330 AOT_hence ‹[λxy ∀F ([F]x ≡ [F]y)]xy›
7331 by (rule "β→C"(1))
7332 AOT_hence ‹∀F ([F]x ≡ [F]y)›
7333 using "β→C"(1) old.prod.case by fast
7334 AOT_hence ‹[A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y)› using 0 "&E" "&I" by blast
7335 AOT_hence ‹∃y ([A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y))› by (rule "∃I")
7336 AOT_thus ‹∃x∃y ([A!]x & [A!]y & x ≠ y & ∀F ([F]x ≡ [F]y))› by (rule "∃I"(2))
7337qed
7338
7339AOT_theorem "kirchner-thm:1": ‹[λx φ{x}]↓ ≡ □∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7340proof(rule "≡I"; rule "→I")
7341 AOT_assume ‹[λx φ{x}]↓›
7342 AOT_hence ‹□[λx φ{x}]↓› by (metis "exist-nec" "vdash-properties:10")
7343 moreover AOT_have ‹□[λx φ{x}]↓ → □∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7344 proof (rule "RM:1"; rule "→I"; rule GEN; rule GEN; rule "→I")
7345 AOT_modally_strict {
7346 fix x y
7347 AOT_assume 0: ‹[λx φ{x}]↓›
7348 moreover AOT_assume ‹∀F([F]x ≡ [F]y)›
7349 ultimately AOT_have ‹[λx φ{x}]x ≡ [λx φ{x}]y›
7350 using "∀E" by blast
7351 AOT_thus ‹(φ{x} ≡ φ{y})›
7352 using "beta-C-meta"[THEN "→E", OF 0] "≡E"(6) by meson
7353 }
7354 qed
7355 ultimately AOT_show ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7356 using "→E" by blast
7357next
7358 AOT_have ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y})) → □∀y(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7359 proof(rule "RM:1"; rule "→I"; rule GEN)
7360 AOT_modally_strict {
7361 AOT_assume ‹∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7362 AOT_hence indisc: ‹φ{x} ≡ φ{y}› if ‹∀F([F]x ≡ [F]y)› for x y
7363 using "∀E"(2) "→E" that by blast
7364 AOT_show ‹(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})› for y
7365 proof (rule "raa-cor:1")
7366 AOT_assume ‹¬(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7367 AOT_hence ‹(∃x(∀F([F]x ≡ [F]y) & φ{x}) & ¬φ{y}) ∨ (¬(∃x(∀F([F]x ≡ [F]y) & φ{x})) & φ{y})›
7368 using "≡E"(1) "oth-class-taut:4:h" by blast
7369 moreover {
7370 AOT_assume 0: ‹∃x(∀F([F]x ≡ [F]y) & φ{x}) & ¬φ{y}›
7371 AOT_obtain a where ‹∀F([F]a ≡ [F]y) & φ{a}›
7372 using "∃E"[rotated, OF 0[THEN "&E"(1)]] by blast
7373 AOT_hence ‹φ{y}› using indisc[THEN "≡E"(1)] "&E" by blast
7374 AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7375 }
7376 moreover {
7377 AOT_assume 0: ‹(¬(∃x(∀F([F]x ≡ [F]y) & φ{x})) & φ{y})›
7378 AOT_hence ‹∀x ¬(∀F([F]x ≡ [F]y) & φ{x})›
7379 using "&E"(1) "cqt-further:4" "→E" by blast
7380 AOT_hence ‹¬(∀F([F]y ≡ [F]y) & φ{y})› using "∀E" by blast
7381 AOT_hence ‹¬∀F([F]y ≡ [F]y) ∨ ¬φ{y}›
7382 using "≡E"(1) "oth-class-taut:5:c" by blast
7383 moreover AOT_have ‹∀F([F]y ≡ [F]y)› by (simp add: "oth-class-taut:3:a" "universal-cor")
7384 ultimately AOT_have ‹¬φ{y}› by (metis "¬¬I" "∨E"(2))
7385 AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7386 }
7387 ultimately AOT_show ‹p & ¬p› for p using "∨E"(3) "raa-cor:1" by blast
7388 qed
7389 }
7390 qed
7391 moreover AOT_assume ‹□∀x∀y(∀F([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7392 ultimately AOT_have ‹□∀y(∃x(∀F([F]x ≡ [F]y) & φ{x}) ≡ φ{y})›
7393 using "→E" by blast
7394 AOT_thus ‹[λx φ{x}]↓›
7395 by (rule "safe-ext"[axiom_inst, THEN "→E", OF "&I", rotated]) "cqt:2[lambda]"
7396qed
7397
7398AOT_theorem "kirchner-thm:2": ‹[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓ ≡ □∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7399proof(rule "≡I"; rule "→I")
7400 AOT_assume ‹[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓›
7401 AOT_hence ‹□[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓› by (metis "exist-nec" "vdash-properties:10")
7402 moreover AOT_have ‹□[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓ → □∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7403 proof (rule "RM:1"; rule "→I"; rule GEN; rule GEN; rule "→I")
7404 AOT_modally_strict {
7405 fix x⇩1x⇩n y⇩1y⇩n :: ‹'a AOT_var›
7406 AOT_assume 0: ‹[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓›
7407 moreover AOT_assume ‹∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)›
7408 ultimately AOT_have ‹[λx⇩1...x⇩n φ{x⇩1...x⇩n}]x⇩1...x⇩n ≡ [λx⇩1...x⇩n φ{x⇩1...x⇩n}]y⇩1...y⇩n›
7409 using "∀E" by blast
7410 AOT_thus ‹(φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n})›
7411 using "beta-C-meta"[THEN "→E", OF 0] "≡E"(6) by meson
7412 }
7413 qed
7414 ultimately AOT_show ‹□∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7415 using "→E" by blast
7416next
7417 AOT_have ‹□(∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))) →
7418 □∀y⇩1...∀y⇩n((∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})) ≡ φ{y⇩1...y⇩n})›
7419 proof(rule "RM:1"; rule "→I"; rule GEN)
7420 AOT_modally_strict {
7421 AOT_assume ‹∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7422 AOT_hence indisc: ‹φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}› if ‹∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)› for x⇩1x⇩n y⇩1y⇩n
7423 using "∀E"(2) "→E" that by blast
7424 AOT_show ‹(∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})) ≡ φ{y⇩1...y⇩n}› for y⇩1y⇩n
7425 proof (rule "raa-cor:1")
7426 AOT_assume ‹¬((∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})) ≡ φ{y⇩1...y⇩n})›
7427 AOT_hence ‹((∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})) & ¬φ{y⇩1...y⇩n}) ∨
7428 (¬(∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})) & φ{y⇩1...y⇩n})›
7429 using "≡E"(1) "oth-class-taut:4:h" by blast
7430 moreover {
7431 AOT_assume 0: ‹(∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})) & ¬φ{y⇩1...y⇩n}›
7432 AOT_obtain a⇩1a⇩n where ‹∀F([F]a⇩1...a⇩n ≡ [F]y⇩1...y⇩n) & φ{a⇩1...a⇩n}›
7433 using "∃E"[rotated, OF 0[THEN "&E"(1)]] by blast
7434 AOT_hence ‹φ{y⇩1...y⇩n}› using indisc[THEN "≡E"(1)] "&E" by blast
7435 AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7436 }
7437 moreover {
7438 AOT_assume 0: ‹(¬((∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n}))) & φ{y⇩1...y⇩n})›
7439 AOT_hence ‹∀x⇩1...∀x⇩n ¬(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})›
7440 using "&E"(1) "cqt-further:4" "→E" by blast
7441 AOT_hence ‹¬(∀F([F]y⇩1...y⇩n ≡ [F]y⇩1...y⇩n) & φ{y⇩1...y⇩n})› using "∀E" by blast
7442 AOT_hence ‹¬∀F([F]y⇩1...y⇩n ≡ [F]y⇩1...y⇩n) ∨ ¬φ{y⇩1...y⇩n}›
7443 using "≡E"(1) "oth-class-taut:5:c" by blast
7444 moreover AOT_have ‹∀F([F]y⇩1...y⇩n ≡ [F]y⇩1...y⇩n)›
7445 by (simp add: "oth-class-taut:3:a" "universal-cor")
7446 ultimately AOT_have ‹¬φ{y⇩1...y⇩n}› by (metis "¬¬I" "∨E"(2))
7447 AOT_hence ‹p & ¬p› for p using 0[THEN "&E"(2)] "&I" "raa-cor:3" by blast
7448 }
7449 ultimately AOT_show ‹p & ¬p› for p using "∨E"(3) "raa-cor:1" by blast
7450 qed
7451 }
7452 qed
7453 moreover AOT_assume ‹□∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7454 ultimately AOT_have ‹□∀y⇩1...∀y⇩n((∃x⇩1...∃x⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) & φ{x⇩1...x⇩n})) ≡ φ{y⇩1...y⇩n})›
7455 using "→E" by blast
7456 AOT_thus ‹[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓›
7457 by (rule "safe-ext"[axiom_inst, THEN "→E", OF "&I", rotated]) "cqt:2[lambda]"
7458qed
7459
7460AOT_theorem "kirchner-thm-cor:1": ‹[λx φ{x}]↓ → ∀x∀y(∀F([F]x ≡ [F]y) → □(φ{x} ≡ φ{y}))›
7461proof(rule "→I"; rule GEN; rule GEN; rule "→I")
7462 fix x y
7463 AOT_assume ‹[λx φ{x}]↓›
7464 AOT_hence ‹□∀x∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7465 by (rule "kirchner-thm:1"[THEN "≡E"(1)])
7466 AOT_hence ‹∀x□∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7467 using CBF[THEN "→E"] by blast
7468 AOT_hence ‹□∀y (∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7469 using "∀E" by blast
7470 AOT_hence ‹∀y □(∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7471 using CBF[THEN "→E"] by blast
7472 AOT_hence ‹□(∀F ([F]x ≡ [F]y) → (φ{x} ≡ φ{y}))›
7473 using "∀E" by blast
7474 AOT_hence ‹□∀F ([F]x ≡ [F]y) → □(φ{x} ≡ φ{y})›
7475 using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7476 moreover AOT_assume ‹∀F([F]x ≡ [F]y)›
7477 ultimately AOT_show ‹□(φ{x} ≡ φ{y})› using "→E" "ind-nec" by blast
7478qed
7479
7480AOT_theorem "kirchner-thm-cor:2":
7481 ‹[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓ → ∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n(∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → □(φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7482proof(rule "→I"; rule GEN; rule GEN; rule "→I")
7483 fix x⇩1x⇩n y⇩1y⇩n
7484 AOT_assume ‹[λx⇩1...x⇩n φ{x⇩1...x⇩n}]↓›
7485 AOT_hence 0: ‹□∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n (∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7486 by (rule "kirchner-thm:2"[THEN "≡E"(1)])
7487 AOT_have ‹∀x⇩1...∀x⇩n∀y⇩1...∀y⇩n □(∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7488 proof(rule GEN; rule GEN)
7489 fix x⇩1x⇩n y⇩1y⇩n
7490 AOT_show ‹□(∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7491 apply (rule "RM:1"[THEN "→E", rotated, OF 0]; rule "→I")
7492 using "∀E" by blast
7493 qed
7494 AOT_hence ‹∀y⇩1...∀y⇩n □(∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7495 using "∀E" by blast
7496 AOT_hence ‹□(∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7497 using "∀E" by blast
7498 AOT_hence ‹□(∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → (φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n}))›
7499 using "∀E" by blast
7500 AOT_hence 0: ‹□∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n) → □(φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n})›
7501 using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7502 moreover AOT_assume ‹∀F([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)›
7503 moreover AOT_have ‹[λx⇩1...x⇩n □∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)]↓› by "cqt:2[lambda]"
7504 ultimately AOT_have ‹[λx⇩1...x⇩n □∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)]x⇩1...x⇩n ≡ [λx⇩1...x⇩n □∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)]y⇩1...y⇩n›
7505 using "∀E" by blast
7506 moreover AOT_have ‹[λx⇩1...x⇩n □∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)]y⇩1...y⇩n›
7507 apply (rule "β←C"(1))
7508 apply "cqt:2[lambda]"
7509 apply (fact "cqt:2[const_var]"[axiom_inst])
7510 by (simp add: RN GEN "oth-class-taut:3:a")
7511 ultimately AOT_have ‹[λx⇩1...x⇩n □∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)]x⇩1...x⇩n› using "≡E"(2) by blast
7512 AOT_hence ‹□∀F ([F]x⇩1...x⇩n ≡ [F]y⇩1...y⇩n)›
7513 using "β→C"(1) by blast
7514 AOT_thus ‹□(φ{x⇩1...x⇩n} ≡ φ{y⇩1...y⇩n})› using "→E" 0 by blast
7515qed
7516
7517AOT_define propositional :: ‹Π ⇒ φ› (‹Propositional'(_')›)
7518 "prop-prop1": ‹Propositional([F]) ≡⇩d⇩f ∃p(F = [λy p])›
7519
7520AOT_theorem "prop-prop2:1": ‹∀p [λy p]↓›
7521 by (rule GEN) "cqt:2[lambda]"
7522
7523AOT_theorem "prop-prop2:2": ‹[λν φ]↓›
7524 by "cqt:2[lambda]"
7525
7526AOT_theorem "prop-prop2:3": ‹F = [λy p] → □∀x([F]x ≡ p)›
7527proof (rule "→I")
7528 AOT_assume 0: ‹F = [λy p]›
7529 AOT_show ‹□∀x([F]x ≡ p)›
7530 by (rule "rule=E"[rotated, OF 0[symmetric]]; rule RN; rule GEN; rule "beta-C-meta"[THEN "→E"])
7531 "cqt:2[lambda]"
7532qed
7533
7534AOT_theorem "prop-prop2:4": ‹Propositional([F]) → □Propositional([F])›
7535proof(rule "→I")
7536 AOT_assume ‹Propositional([F])›
7537 AOT_hence ‹∃p(F = [λy p])› using "≡⇩d⇩fE"[OF "prop-prop1"] by blast
7538 then AOT_obtain p where ‹F = [λy p]› using "∃E"[rotated] by blast
7539 AOT_hence ‹□(F = [λy p])› using "id-nec:2" "modus-tollens:1" "raa-cor:3" by blast
7540 AOT_hence ‹∃p □(F = [λy p])› using "∃I" by fast
7541 AOT_hence 0: ‹□∃p (F = [λy p])› by (metis Buridan "vdash-properties:10")
7542 AOT_thus ‹□Propositional([F])›
7543 using "prop-prop1"[THEN "≡Df"]
7544 by (AOT_subst ‹Propositional([F])› ‹∃p (F = [λy p])›) auto
7545qed
7546
7547AOT_define indicriminate :: ‹Π ⇒ φ› ("Indiscriminate'(_')")
7548 "prop-indis": ‹Indiscriminate([F]) ≡⇩d⇩f F↓ & □(∃x [F]x → ∀x [F]x)›
7549
7550AOT_theorem "prop-in-thm": ‹Propositional([Π]) → Indiscriminate([Π])›
7551proof(rule "→I")
7552 AOT_assume ‹Propositional([Π])›
7553 AOT_hence ‹∃p Π = [λy p]› using "≡⇩d⇩fE"[OF "prop-prop1"] by blast
7554 then AOT_obtain p where Π_def: ‹Π = [λy p]› using "∃E"[rotated] by blast
7555 AOT_show ‹Indiscriminate([Π])›
7556 proof (rule "≡⇩d⇩fI"[OF "prop-indis"]; rule "&I")
7557 AOT_show ‹Π↓›
7558 using Π_def by (meson "t=t-proper:1" "vdash-properties:6")
7559 next
7560 AOT_show ‹□(∃x [Π]x → ∀x [Π]x)›
7561 proof (rule "rule=E"[rotated, OF Π_def[symmetric]]; rule RN; rule "→I"; rule GEN)
7562 AOT_modally_strict {
7563 AOT_assume ‹∃x [λy p]x›
7564 then AOT_obtain a where ‹[λy p]a› using "∃E"[rotated] by blast
7565 AOT_hence 0: ‹p› by (metis "β→C"(1))
7566 AOT_show ‹[λy p]x› for x
7567 apply (rule "β←C"(1))
7568 apply "cqt:2[lambda]"
7569 apply (fact "cqt:2[const_var]"[axiom_inst])
7570 by (fact 0)
7571 }
7572 qed
7573 qed
7574qed
7575
7576AOT_theorem "prop-in-f:1": ‹Necessary([F]) → Indiscriminate([F])›
7577proof (rule "→I")
7578 AOT_assume ‹Necessary([F])›
7579 AOT_hence 0: ‹□∀x⇩1...∀x⇩n [F]x⇩1...x⇩n› using "≡⇩d⇩fE"[OF "contingent-properties:1"] by blast
7580 AOT_show ‹Indiscriminate([F])›
7581 by (rule "≡⇩d⇩fI"[OF "prop-indis"])
7582 (metis "0" "KBasic:1" "&I" "ex:1:a" "rule-ui:2[const_var]" "vdash-properties:6")
7583qed
7584
7585AOT_theorem "prop-in-f:2": ‹Impossible([F]) → Indiscriminate([F])›
7586proof (rule "→I")
7587 AOT_modally_strict {
7588 AOT_have ‹∀x ¬[F]x → (∃x [F]x → ∀x [F]x)›
7589 by (metis "instantiation" "cqt-orig:3" "Hypothetical Syllogism" "deduction-theorem" "raa-cor:3")
7590 }
7591 AOT_hence 0: ‹□∀x ¬[F]x → □(∃x [F]x → ∀x [F]x)›
7592 by (rule "RM:1")
7593 AOT_assume ‹Impossible([F])›
7594 AOT_hence ‹□∀x ¬[F]x› using "≡⇩d⇩fE"[OF "contingent-properties:2"] "&E" by blast
7595 AOT_hence 1: ‹□(∃x [F]x → ∀x [F]x)› using 0 "→E" by blast
7596 AOT_show ‹Indiscriminate([F])›
7597 by (rule "≡⇩d⇩fI"[OF "prop-indis"]; rule "&I")
7598 (simp add: "ex:1:a" "rule-ui:2[const_var]" 1)+
7599qed
7600
7601AOT_theorem "prop-in-f:3:a": ‹¬Indiscriminate([E!])›
7602proof(rule "raa-cor:2")
7603 AOT_assume ‹Indiscriminate([E!])›
7604 AOT_hence 0: ‹□(∃x [E!]x → ∀x [E!]x)›
7605 using "≡⇩d⇩fE"[OF "prop-indis"] "&E" by blast
7606 AOT_hence ‹◇∃x [E!]x → ◇∀x [E!]x›
7607 using "KBasic:13" "vdash-properties:10" by blast
7608 moreover AOT_have ‹◇∃x [E!]x›
7609 by (simp add: "thm-cont-e:3")
7610 ultimately AOT_have ‹◇∀x [E!]x›
7611 by (metis "vdash-properties:6")
7612 AOT_thus ‹p & ¬p› for p
7613 by (metis "≡⇩d⇩fE" "conventions:5" "o-objects-exist:5" "reductio-aa:1")
7614qed
7615
7616AOT_theorem "prop-in-f:3:b": ‹¬Indiscriminate([E!]⇧-)›
7617proof (rule "rule=E"[rotated, OF "rel-neg-T:2"[symmetric]]; rule "raa-cor:2")
7618 AOT_assume ‹Indiscriminate([λx ¬[E!]x])›
7619 AOT_hence 0: ‹□(∃x [λx ¬[E!]x]x → ∀x [λx ¬[E!]x]x)›
7620 using "≡⇩d⇩fE"[OF "prop-indis"] "&E" by blast
7621 AOT_hence ‹□∃x [λx ¬[E!]x]x → □∀x [λx ¬[E!]x]x›
7622 using "→E" "qml:1" "vdash-properties:1[2]" by blast
7623 moreover AOT_have ‹□∃x [λx ¬[E!]x]x›
7624 apply (AOT_subst ‹[λx ¬E!x]x› ‹¬E!x› bound: x)
7625 apply (rule "beta-C-meta"[THEN "→E"])
7626 apply "cqt:2[lambda]"
7627 by (metis (full_types) "B◇" RN "T◇" "cqt-further:2" "o-objects-exist:5" "vdash-properties:10")
7628 ultimately AOT_have 1: ‹□∀x [λx ¬[E!]x]x›
7629 by (metis "vdash-properties:6")
7630 AOT_hence ‹□∀x ¬[E!]x›
7631 by (AOT_subst (reverse) ‹¬[E!]x› ‹[λx ¬[E!]x]x› bound: x)
7632 (auto intro!: "cqt:2" "beta-C-meta"[THEN "→E"])
7633 AOT_hence ‹∀x □¬[E!]x› by (metis "CBF" "vdash-properties:10")
7634 moreover AOT_obtain a where abs_a: ‹O!a›
7635 using "instantiation" "o-objects-exist:1" "qml:2" "vdash-properties:1[2]" "vdash-properties:6" by blast
7636 ultimately AOT_have ‹□¬[E!]a› using "∀E" by blast
7637 AOT_hence 2: ‹¬◇[E!]a› by (metis "≡⇩d⇩fE" "conventions:5" "reductio-aa:1")
7638 AOT_have ‹A!a›
7639 apply (rule "=⇩d⇩fI"(2)[OF AOT_abstract])
7640 apply "cqt:2[lambda]"
7641 apply (rule "β←C"(1))
7642 apply "cqt:2[lambda]"
7643 using "cqt:2[const_var]"[axiom_inst] apply blast
7644 by (fact 2)
7645 AOT_thus ‹p & ¬p› for p using abs_a
7646 by (metis "≡E"(1) "oa-contingent:2" "reductio-aa:1")
7647qed
7648
7649AOT_theorem "prop-in-f:3:c": ‹¬Indiscriminate(O!)›
7650proof(rule "raa-cor:2")
7651 AOT_assume ‹Indiscriminate(O!)›
7652 AOT_hence 0: ‹□(∃x O!x → ∀x O!x)›
7653 using "≡⇩d⇩fE"[OF "prop-indis"] "&E" by blast
7654 AOT_hence ‹□∃x O!x → □∀x O!x›
7655 using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7656 moreover AOT_have ‹□∃x O!x›
7657 using "o-objects-exist:1" by blast
7658 ultimately AOT_have ‹□∀x O!x›
7659 by (metis "vdash-properties:6")
7660 AOT_thus ‹p & ¬p› for p
7661 by (metis "o-objects-exist:3" "qml:2" "raa-cor:3" "vdash-properties:10" "vdash-properties:1[2]")
7662qed
7663
7664AOT_theorem "prop-in-f:3:d": ‹¬Indiscriminate(A!)›
7665proof(rule "raa-cor:2")
7666 AOT_assume ‹Indiscriminate(A!)›
7667 AOT_hence 0: ‹□(∃x A!x → ∀x A!x)›
7668 using "≡⇩d⇩fE"[OF "prop-indis"] "&E" by blast
7669 AOT_hence ‹□∃x A!x → □∀x A!x›
7670 using "qml:1"[axiom_inst] "vdash-properties:6" by blast
7671 moreover AOT_have ‹□∃x A!x›
7672 using "o-objects-exist:2" by blast
7673 ultimately AOT_have ‹□∀x A!x›
7674 by (metis "vdash-properties:6")
7675 AOT_thus ‹p & ¬p› for p
7676 by (metis "o-objects-exist:4" "qml:2" "raa-cor:3" "vdash-properties:10" "vdash-properties:1[2]")
7677qed
7678
7679AOT_theorem "prop-in-f:4:a": ‹¬Propositional(E!)›
7680 using "modus-tollens:1" "prop-in-f:3:a" "prop-in-thm" by blast
7681
7682AOT_theorem "prop-in-f:4:b": ‹¬Propositional(E!⇧-)›
7683 using "modus-tollens:1" "prop-in-f:3:b" "prop-in-thm" by blast
7684
7685AOT_theorem "prop-in-f:4:c": ‹¬Propositional(O!)›
7686 using "modus-tollens:1" "prop-in-f:3:c" "prop-in-thm" by blast
7687
7688AOT_theorem "prop-in-f:4:d": ‹¬Propositional(A!)›
7689 using "modus-tollens:1" "prop-in-f:3:d" "prop-in-thm" by blast
7690
7691AOT_theorem "prop-prop-nec:1": ‹◇∃p (F = [λy p]) → ∃p(F = [λy p])›
7692proof(rule "→I")
7693 AOT_assume ‹◇∃p (F = [λy p])›
7694 AOT_hence ‹∃p ◇(F = [λy p])›
7695 by (metis "BF◇" "vdash-properties:10")
7696 then AOT_obtain p where ‹◇(F = [λy p])› using "∃E"[rotated] by blast
7697 AOT_hence ‹F = [λy p]› by (metis "derived-S5-rules:2" emptyE "id-nec:2" "vdash-properties:6")
7698 AOT_thus ‹∃p(F = [λy p])› by (rule "∃I")
7699qed
7700
7701AOT_theorem "prop-prop-nec:2": ‹∀p (F ≠ [λy p]) → □∀p(F ≠ [λy p])›
7702proof(rule "→I")
7703 AOT_assume ‹∀p (F ≠ [λy p])›
7704 AOT_hence ‹(F ≠ [λy p])› for p
7705 using "∀E" by blast
7706 AOT_hence ‹□(F ≠ [λy p])› for p
7707 by (rule "id-nec2:2"[unvarify β, THEN "→E", rotated]) "cqt:2[lambda]"
7708 AOT_hence ‹∀p □(F ≠ [λy p])› by (rule GEN)
7709 AOT_thus ‹□∀p (F ≠ [λy p])› using BF[THEN "→E"] by fast
7710qed
7711
7712AOT_theorem "prop-prop-nec:3": ‹∃p (F = [λy p]) → □∃p(F = [λy p])›
7713proof(rule "→I")
7714 AOT_assume ‹∃p (F = [λy p])›
7715 then AOT_obtain p where ‹(F = [λy p])› using "∃E"[rotated] by blast
7716 AOT_hence ‹□(F = [λy p])› by (metis "id-nec:2" "vdash-properties:6")
7717 AOT_hence ‹∃p□(F = [λy p])› by (rule "∃I")
7718 AOT_thus ‹□∃p(F = [λy p])› by (metis Buridan "vdash-properties:10")
7719qed
7720
7721AOT_theorem "prop-prop-nec:4": ‹◇∀p (F ≠ [λy p]) → ∀p(F ≠ [λy p])›
7722proof(rule "→I")
7723 AOT_assume ‹◇∀p (F ≠ [λy p])›
7724 AOT_hence ‹∀p ◇(F ≠ [λy p])› by (metis "Buridan◇" "vdash-properties:10")
7725 AOT_hence ‹◇(F ≠ [λy p])› for p
7726 using "∀E" by blast
7727 AOT_hence ‹F ≠ [λy p]› for p
7728 by (rule "id-nec2:3"[unvarify β, THEN "→E", rotated]) "cqt:2[lambda]"
7729 AOT_thus ‹∀p (F ≠ [λy p])› by (rule GEN)
7730qed
7731
7732AOT_theorem "enc-prop-nec:1": ‹◇∀F (x[F] → ∃p(F = [λy p])) → ∀F(x[F] → ∃p (F = [λy p]))›
7733proof(rule "→I"; rule GEN; rule "→I")
7734 fix F
7735 AOT_assume ‹◇∀F (x[F] → ∃p(F = [λy p]))›
7736 AOT_hence ‹∀F ◇(x[F] → ∃p(F = [λy p]))›
7737 using "Buridan◇" "vdash-properties:10" by blast
7738 AOT_hence 0: ‹◇(x[F] → ∃p(F = [λy p]))› using "∀E" by blast
7739 AOT_assume ‹x[F]›
7740 AOT_hence ‹□x[F]› by (metis "en-eq:2[1]" "≡E"(1))
7741 AOT_hence ‹◇∃p(F = [λy p])›
7742 using 0 by (metis "KBasic2:4" "≡E"(1) "vdash-properties:10")
7743 AOT_thus ‹∃p(F = [λy p])›
7744 using "prop-prop-nec:1"[THEN "→E"] by blast
7745qed
7746
7747AOT_theorem "enc-prop-nec:2": ‹∀F (x[F] → ∃p(F = [λy p])) → □∀F(x[F] → ∃p (F = [λy p]))›
7748 using "derived-S5-rules:1"[where Γ="{}", simplified, OF "enc-prop-nec:1"]
7749 by blast
7750
7751
7752end
7753